Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic meters

Answer: 150.72 cubic meters

1. Given that the base radius of the cone is 4 meters and the height is 9 meters.

2. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the base radius and h is the height.

3. Substitute the given radius r = 4 meters and height h = 9 meters into the cone volume formula, yielding V = (1/3) * π * (4^2) * 9.

4. Calculate the square of 4, which is 16, and substitute it into the formula to get V = (1/3) * π * 16 * 9.

5. Calculate 16 * 9 = 144, thus V = (1/3) * π * 144.

6. Take π ≈ 3.14 and substitute it into the formula, yielding V ≈ (1/3) * 3.14 * 144.

7. Calculate 3.14 * 144 = 452.16.

8. Further calculate V ≈ (1/3) * 452.16 = 150.72.

9. Therefore, the volume of the cone is approximately 150.72 cubic meters.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

<note>A cone refers to a given geometric figure, with a base that is a circle with a radius of 4 meters and a height of 9 meters</note>.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

In the diagram of this problem, <note>the base of the cone is a circle</note>, <note>the radius of the circle is 4 meters</note>. All points in the diagram <note>that are 4 meters away from the center of the circle are on the circle</note>.

Volume Formula of a Cone

The formula to calculate the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base circle and \( h \) is the height of the cone.

Original: <note>Volume Formula of a Cone V = (1/3) * π * r^2 * h</note>. Given <note>the radius of the base circle r = 4 meters</note>, <note>height h = 9 meters</note>, substituting these values into the formula gives <note>V = (1/3) * π * (4^2) * 9</note>, further resulting in a volume of approximately <note>150.72 cubic meters</note>.


Question: What is the volume? $\Box$  cubic inches

Answer: 70 cubic inches

1. First, identify that the base of the triangular prism is a triangle with a base length of 7 inches, a height of 5 inches, and the length of the prism is 4 inches.

2. Calculate the area of the triangle using the area formula for a triangle: Area = 1/2 × base × height.

3. Substitute the given base length and height: base length is 7 inches, height is 5 inches. Therefore, the area of the triangle is 1/2 × 7 × 5 = 17.5 square inches.

4. The volume formula for a prism is: Volume of the triangular prism = base area × length.

5. Substitute the calculated base area of 17.5 square inches and the length of 4 inches into the volume formula:

6. The volume of the triangular prism is 17.5 × 4 = 70 cubic inches.

7. Through the above reasoning, the final answer is 70 cubic inches.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

The <note>triangle</note> is the base of the triangular prism. The <note>triangle</note> is composed of three <note>non-collinear points</note> and their <note>connecting line segments</note>, with the <note>base</note> length being 7 inches and the <note>height</note> being 5 inches.

Definition of Triangular Prism

A triangular prism is a type of hexahedron that is formed by two parallel and congruent triangular bases and three rectangular lateral faces.

<note>In the diagram of this problem, the base of the triangular prism consists of equal and parallel triangles</note>, with a <note>base length of 7 inches</note>, <note>height of 5 inches</note>, and <note>length of the prism is 4 inches</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

<note>The side length of 7 inches is the base</note>, <note>The segment length of 5 inches is the height on that base</note>, so the area of the triangle is equal to the base of 7 inches multiplied by the height of 5 inches divided by 2, that is, Area = (7 * 5) / 2.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

<note>In the given figure</note>, in the prism, <note>the area of the base is 17.5</note>, <note>the height of the prism is 4</note>. Therefore, according to the volume formula of the prism, the volume of the prism is equal to <note>base area 17.5 multiplied by height 4</note>.


Question: What is the volume of this rectangular pyramid? $\Box$  cubic millimeters

Answer: 300 cubic millimeters

1. Observe the given pyramid, the base is a square with a side length of 10 mm.

2. Calculate the area of this square base. The formula for the area of a square is the square of the side length, so the area is $10 	imes 10 = 100$ square millimeters.

3. According to the information given in the problem, the height of the pyramid is 9 mm.

4. Calculate its volume using the pyramid volume formula $V = rac{1}{3} 	imes 	ext{base area} 	imes 	ext{height}$. Substitute the known information into the formula, we have $V = rac{1}{3} 	imes 100 	imes 9$.

5. Perform the calculation to get $V = rac{1}{3} 	imes 900 = 300$ cubic millimeters.

6. Through the above reasoning, the final answer is 300 cubic millimeters.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

In the figure of this problem, <note>the side length of the base of the quadrilateral is 10 mm</note>, and <note>all four interior angles are 90 degrees</note>, so the base is a square.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

Original text: <note>Side length a is 10 mm</note>, so the area of the square base is <note>Area A = 10 x 10 = 100 square millimeters</note>.

Volume Formula of Pyramid

The volume \( V \) of a pyramid is equal to one third of the product of its base area and its height. Mathematically, this is expressed as: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

<note>The base area A is 100 square millimeters</note>, <note>the height is 9 millimeters</note>, so <note>the volume V = 1/3 x 100 x 9 = 300 cubic millimeters</note>.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic millimeters

Answer: 235.50 cubic millimeters

1. Observe the given geometric figure, which is a cone, and the radius of the base and the height of the cone are given.

2. According to the information in the figure, the radius of the base of the cone is 5 mm and the height is 9 mm.

3. The volume formula of the cone is V = (1/3) * π * r^2 * h.

4. Substitute the known numbers into the volume formula to get V = (1/3) * 3.14 * (5)^2 * 9.

5. Calculate (5)^2 = 25.

6. Substitute the values to get V = (1/3) * 3.14 * 25 * 9.

7. Calculate 25 * 9 = 225.

8. Then calculate 3.14 * 225 = 706.5.

9. Calculate (1/3) * 706.5 = 235.5

10. Through the above reasoning, the final answer is 235.50 cubic millimeters.

Definition of Cone

A <note>cone</note>, with a <note>base that is a circle with a radius of r=5 mm</note>, and a <note>vertex height of h=9 mm</note>.

Volume Formula of a Cone

Use the <note>formula V = (1/3) * π * r^2 * h</note> to calculate the volume of the cone, given <note>base radius r=5 mm</note>, <note>height h=9 mm</note>. Therefore, substituting into the formula gives <note>V = (1/3) * 3.14 * (5)^2 * 9</note>.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic millimeters

Answer: 2,143.57 cubic millimeters

1. Given the radius of the sphere is 8 mm, according to the sphere volume formula, the sphere volume formula is V = (4/3)πr³.

2. Substitute the radius r = 8 mm into the sphere volume formula, obtaining V = (4/3)π(8 mm)³.

3. Calculate the cube of 8 mm: (8 mm)³ = 8 × 8 × 8 = 512 mm³.

4. Substitute the above value into the volume expression, getting V = (4/3)π × 512 mm³.

5. Calculate the value of (4/3)π × 512 mm³, first calculate (4/3) × 512 ≈ 682.67.

6. Using the given approximate value π ≈ 3.14, obtain V ≈ 682.67 × 3.14.

7. Calculate 682.67 × 3.14 ≈ 2143.5838.

8. Approximate 2143.5838 to the nearest two decimal places, the result is 2143.58.

9. Through the above reasoning, the final answer is 2143.58 cubic millimeters.

Formula for the Volume of a Sphere

The volume \(V\) of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius of the sphere.

In this problem, <note>the radius of the sphere \( r = 8 \text{ mm} \)</note>, substituting <note> \( r \) into the formula</note> gives <note> \( V = \left( \frac{4}{3} \right) \pi (8 \text{ mm})^3 \)</note>, used for <note>calculating the volume of the sphere</note>.


Question:
In circle A, $\overset{\frown}{BDC}$ is highlighted.  $\angle $ BAC measures 45°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{BDC}$ in centimeters?

- 2/3(3𝜋)
- 7/8(9𝜋)
- 7/8(3𝜋)
- 2/3(9𝜋)

Answer:
7/8

 7/8(3𝜋)

1. In circle A, arc BDC is the highlighted part, given that angle BAC is 45°.

2. According to the property of central angles, the arc corresponding to angle BAC is also 45°, which is arc BC.

3. The circumference of the circle is 360°, representing the entire circle.

4. The degree measure of arc BC is 45/360 of the entire circle.

5. Simplifying 45/360, we get 1/8. Therefore, 1/8 of the circle is not highlighted.

6. Since arc BDC is highlighted, the portion of the circle it occupies should be 1 - 1/8 = 7/8.

7. The formula for the circumference of a circle is C = πd, where d is the diameter.

8. The problem states that the diameter d = 3 cm.

9. Calculate the circumference of the circle C = π * 3 = 3π.

10. Since arc BDC is 7/8 of the circumference, the length of arc BDC is (7/8) * 3π.

11. From the above calculation, the length of arc BDC is 7/8 * 3π.

Definition of Circle

In circle A, point A is the center, the diameter is 3 cm, the radius is 3/2 cm. In the figure, all points that are at a distance of 3/2 cm from point A are on circle A.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

In the diagram of this problem, there are three points <note>B</note>, <note>D</note>, and <note>C</note> on the circle <note>A</note>. The arc <note>BC</note> is a segment of the curve connecting the two points <note>B</note> and <note>C</note>. The arc <note>BDC</note> is a segment of the curve connecting the three points <note>B</note>, <note>D</note>, and <note>C</note>. According to the definition of an arc, the arc <note>BDC</note> is a segment of the curve between the <note>three points B, D, and C</note> on the circle.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

In the diagram of this problem, <note>circle A</note> has <note>point B</note> and <note>point C</note> as two points on the circle, with <note>the center being point A</note>. The angle ∠BAC formed by <note>line segments AB and AC</note> is called the central angle.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>In circle A, point A is the center of the circle</note>, <note>line segment AB is the radius r</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, that is, C = 2πr. <note>Diameter d = 3 cm</note>, therefore <note>radius r = 3/2 cm</note>, the circumference C = 2π * (3/2) = 3π cm.

Arc Length Formula of a Circle

The arc length refers to the length of a segment of the circumference of a circle. It can be calculated using the formula 𝓁 = C × (m/360), where m is the measure of the central angle in degrees.

The original text: <note>Arc BC occupies 45/360 degrees of the circle</note>, simplified to <note>1/8</note>. Therefore, arc BDC occupies the remaining part, which is <note>7/8</note>, so the length of arc BDC is <note>7/8 * 3π</note>.


Question: What is the volume of this rectangular pyramid? $\Box$  cubic millimeters

Answer: 8 cubic millimeters

1. First, identify that the base shape in the given figure is a rectangle with a length of 4 mm and a width of 2 mm.

2. Using the area formula for a rectangle A = length × width, calculate the base area A = 4 mm × 2 mm = 8 mm².

3. Determine the height of the pyramid, which is given as 3 mm in the figure.

4. According to the volume formula for a pyramid V = (1/3) × base area × height, substitute the known values V = (1/3) × 8 mm² × 3 mm.

5. Calculate the volume V = (1/3) × 24 mm³ = 8 mm³.

6. Through the above reasoning, the final answer is 8 mm³.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

<note>The base is a rectangle</note>, its <note>length is 4 millimeters</note>, <note>width is 2 millimeters</note>. The rectangle's <note>four sides are respectively the 4 millimeter side as the length, two sides parallel and equal</note>; <note>the 2 millimeter side as the width, two sides parallel and equal</note>. <note>Each angle is 90 degrees</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

<note>Formula for the Area of a Rectangle</note> in this problem diagram, in the rectangle at the base, <note>side 4 mm and side 2 mm are the length and width of the rectangle</note>, so <note>Area of the rectangle = 4 mm * 2 mm</note>.

Volume Formula of Pyramid

In the figure of this problem, <note>the base area of the pyramid is 8 square millimeters</note>, <note>the height of the pyramid is 3mm</note>. According to the <note>Volume Formula of Pyramid</note>, <note>the volume V of the pyramid is equal to the base area multiplied by the height and then multiplied by 1/3</note>, that is: V = 1/3 * 8 * 3.


Question: What is the volume? $\Box$  cubic yards

Answer: 280 cubic yards

1. First, identify the geometric figure as a triangular prism, and use the volume formula for prisms. The volume formula for a triangular prism is: Volume = Base Area × Height.

2. To calculate the base area of the triangular prism, we need to calculate the area of the base, which is a triangle.

3. The base is a triangle with a base length of 8 yd and a height of 7 yd. The known formula for the area of a triangle is: Area = 1/2 × Base × Height.

4. Substitute the known base length of 8 yd and height of 7 yd into the triangle area formula, and calculate: Area = 1/2 × 8 × 7 = 28 square yd.

5. Determine the height of the triangular prism. According to the problem, the height of the triangular prism is 10 yd.

6. Substitute the base area (28 square yd) and height (10 yd) into the volume formula: Volume = Base Area × Height = 28 × 10 = 280 cubic yd.

7. Through the above reasoning, the final answer is 280 cubic yd.

Definition of Triangular Prism

In the figure of this problem, the geometric body is a <note>triangular prism</note>. The <note>two congruent and parallel triangular bases</note> are respectively the bases of the triangular prism, and the other three faces are <note>rectangles</note>. The <note>side length of the triangular base is 8 yards</note>, the <note>height of the base is 7 yards</note>, and the <note>height of the triangular prism is 10 yards</note>.

Area Formula of a Triangle

<note>The base of the triangular prism is a triangle</note>, <note>the base is 8 yd</note>, <note>the height is 7 yd</note>, so the area of the triangle is equal to the base multiplied by the height divided by 2, i.e., area = (8 * 7) / 2.

Volume Formula of Prism

In the figure of this problem, <note>the area of the base of the prism is 28 square yards</note>, <note>the height of the prism is 10 yards</note>. Therefore, according to the <note>volume formula of the prism</note>, the volume of the prism is equal to <note>the base area multiplied by the height</note>.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic centimeters

Answer: 261.67 cubic centimeters

1. According to the provided cone figure, we can see that the height of this cone is 10 cm and the radius of the base is 5 cm.

2. The volume formula of the cone is 𝑉= (1/3)πr²h. Here, 𝑟 is the radius of the base and h is the height of the cone.

3. Substitute the known radius 𝑟=5 cm and height h=10 cm into the volume formula, we get 𝑉= (1/3)π(5)²(10).

4. First, calculate the square of the radius part, 5^2 = 25.

5. Substitute the square result into the formula and calculate: V = (1/3)π(25)(10) = (1/3)π(250).

6. Since π ≈ 3.14, so V = (1/3)(3.14)(250) = (1/3)(785).

7. Calculate (1/3) * 785 ≈ 261.67.

8. According to the calculation steps, the volume of the cone is approximately 261.67 cubic centimeters.

9. Through the above reasoning, the final answer is 261.67 cubic centimeters.

Volume Formula of a Cone

In this problem, <note>the radius of the base circle r = 5 cm</note>, <note>the height of the cone h = 10 cm</note>. According to this definition, we substitute these values to calculate the volume: V = (1/3)π(5)²(10).

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

The <note>radius of the base circle of the cone</note> is the length of the <note>line segment from the center of the circle to any point on the circumference</note>, therefore the <note>radius r of the base circle of the cone</note> = 5 cm.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic centimeters

Answer: 1,436.03 cubic centimeters

1. According to the conditions given in the problem, the radius of the sphere is 7 cm.

2. To calculate the volume of the sphere, we use the sphere volume formula: Volume = (4/3) * 𝜋 * r^3, where r is the radius of the sphere.

3. Substitute the known radius r = 7 cm into the sphere volume formula, and calculate the volume = (4/3) * 𝜋 * (7)^3.

4. Simplify the calculation: (7)^3 = 343, thus the volume = (4/3) * 𝜋 * 343.

5. According to the problem requirements, use 𝜋 ≈ 3.14 for the calculation, obtaining the volume = (4/3) * 3.14 * 343.

6. Calculate: 4 * 3.14 * 343 = 4308.08.

7. Calculate: (1/3) * 4308.08 = 1436.0266...

8. According to rounding rules, round the calculation result to the nearest hundredth, obtaining the final answer of 1436.03 cubic centimeters.

9. Through the above reasoning, the final answer is 1436.03 cubic centimeters.

Definition of Sphere

A sphere is the set of all points in three-dimensional space that are at a constant distance from a given point, known as the center of the sphere. This constant distance is called the radius of the sphere.

The length of the line segment from the <note>center of the sphere</note> to the <note>surface of the sphere</note> is <note>7 centimeters</note>.

Formula for the Volume of a Sphere

<note>The radius of the sphere r is 7 cm</note>. According to the volume formula, Volume = (4/3) * π * r^3, substituting <note>r = 7 cm</note>, the calculated volume is (4/3) * 3.14 * (7)^3 = 1436.03 cubic centimeters.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic yards

Answer: 133.97 cubic yards

1. Given that the base of the cone is a circle with a radius of 4 yards, and the height of the cone is also 8 yards.

2. According to the cone volume formula V = (1/3)πr^2h, substitute the radius r = 4 and height h = 8.

3. Calculate the base area A = πr^2 = π * (4^2) = 16π.

4. Calculate the volume V = (1/3)πr^2h = (1/3) * 16π * 8 = (1/3) * 128π.

5. Using the given π ≈ 3.14, calculate: V ≈ (1/3) * 128 * 3.14.

6. The calculation gives: V ≈ 133.97333333.

7. Round the result to two decimal places, getting V ≈ 133.97 cubic yards.

8. Through the above reasoning, the final answer is approximately 133.97 cubic yards.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

The <note>base of the cone</note> is a <note>circle with a radius of 4 yards</note>, and the <note>height of the cone is 8 yards</note>. The vertex is located <note>directly above the center of the base circle</note>, and it forms an 8-yard height with <note>a point on the base</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

In the figure of this problem, the base of the cone is a <note>circle with a radius of 4 yards</note>. Therefore, the base area A can be calculated as <note>A = π * (4^2) = 16π</note>.

Volume Formula of a Cone

The formula to calculate the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base circle and \( h \) is the height of the cone.

In the figure of this problem, <note>the radius r of the cone is 4 yards</note>, <note>the height h is also 8 yards</note>. Substitute into the formula: <note>V = (1/3)πr^2h = (1/3) * π * (4^2) * 8 = (1/3) * 128π</note>.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic inches

Answer: 4.19 cubic inches

1. First, it is known that the radius of the sphere is 1 inch.

2. The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius.

3. Substitute the known radius r = 1 inch into the volume formula: V = (4/3)π(1)³.

4. Calculate (1)³, which gives 1.

5. Then calculate (4/3)π × 1 = (4/3)π.

6. Use π ≈ 3.14 for further calculation.

7. Calculate (4/3) × 3.14 ≈ 4.1866667.

8. Round the result to two decimal places, giving 4.19.

9. Through the above reasoning, the final volume of the sphere is approximately 4.19 cubic inches.

Definition of Sphere

<note>The center of the sphere in the definition of the sphere is the point in the diagram</note>, the radius of the sphere is <note>1 inch</note>, and the points on the surface of the sphere are no more than 1 inch away from the center.

Definition of Radius

<note>The radius of the sphere is 1 inch</note>, shown as <note>a line segment from the center of the sphere to a point on the surface of the sphere</note>.

Formula for the Volume of a Sphere

In the problem diagram, <note>Formula for the Volume of a Sphere</note> is applied to the <note>sphere</note> in the diagram, given <note>radius r = 1 inch</note>. Therefore, <note>volume V = (4/3)π(1)³</note>.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic inches

Answer: 37.68 cubic inches

1. Given that the axial section of the cone is a right triangle, where the hypotenuse is the slant height of the cone, and the vertical segment is the height of the cone. It is known from the figure that the slant height is 4 inches and the diameter of the base is 6 inches. Therefore, the radius of the cone is 6/2=3 inches.

2. Since the axial section of the cone is a right triangle, the slant height, the height of the cone, and the radius form a right triangle. Therefore, the Pythagorean theorem can be used to calculate the height of the cone. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

3. Substitute the known values into the Pythagorean theorem formula: slant height squared = radius squared + height squared, i.e., 4^2 = 3^2 + h^2

4. Calculate: 16 = 9 + h^2, therefore h^2 = 16 - 9 = 7

5. Find the square root of the height: h = sqrt(7)

6. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the radius and h is the height.

7. Substitute the determined values into the volume formula: V = (1/3) * 3.14 * (3)^2 * sqrt(7)

8. Calculate V = (1/3) * 3.14 * 9 * sqrt(7) ≈ 37.68 cubic inches.

9. Through the above reasoning, the final answer is 37.68 cubic inches.

Definition of Cone

In the figure of this problem, <note>the diameter of the base of the cone is 6 inches</note>, it can be determined that <note>the radius is 3 inches</note>. <note>The slant height of the cone is 4 inches</note>. <note>The height of the cone is the vertical distance from the top to the base</note>, we need to find this <note>height h</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

<note>The axial cross-section of the cone forms a right triangle</note>, in which <note>the hypotenuse is the slant height of the cone (4 inches)</note>, <note>one leg is the radius (3 inches)</note>, and the other leg is the height h of the cone. According to the Pythagorean Theorem, <note>4^2 = 3^2 + h^2</note>.

Volume Formula of a Cone

<note>The radius r of the cone is 3 inches</note>, the slant height is 4 inches, and we use the Pythagorean theorem to find <note>the height h of the cone = sqrt(7)</note>. Substituting these values into the cone volume formula, we get V = (1/3) * 3.14 * (3)^2 * sqrt(7) ≈ 37.68 cubic inches.


Question: What is the volume of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic meters

Answer: 12.56 cubic meters

1. Given the radius of the base of the cylinder is 2 meters, and the height is 1 meter.

2. The volume of the cylinder can be calculated using the formula V = 𝜋r²h, where r is the radius of the base and h is the height.

3. Substituting the given values, the formula becomes V = 𝜋 × (2^2) × 1.

4. Calculate the area of the base of the cylinder, i.e., 2^2 = 4.

5. According to the formula V = 𝜋 × 4 × 1, calculate 4 multiplied by 1, which equals 4.

7. After the above calculations, the volume of the cylinder is approximately 12.56 cubic meters.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>In the figure of this problem, the cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are two identical circles, <note>each with a radius of 2 meters</note>, and <note>their centers lie on the same straight line</note>. The lateral surface is a rectangle, when unfolded, <note>its height equals the cylinder's height of 1 meter</note>, <note>and its width equals the circumference of the circle</note>.

Definition of Radius

In the diagram of this problem, the radius of the cylinder's base is 2 meters. <note>The length of the line segment from the center of the cylinder's base to any point on the circumference is 2 meters</note>, therefore this line segment is the radius of the circle.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

<note>Radius r = 2 meters</note> and <note>Height h = 1 meter</note>, we substitute into the formula V = 𝜋 × <note>(2^2) × 1</note>, obtaining the volume of the cylinder as 12.56 cubic meters (taking 𝜋 ≈ 3.14).

Area Formula of a Circle

<note>The base of the cylinder is a circle</note>, <note>the radius of the circle is 2 meters</note>, according to the area formula of a circle, <note>the area of the circle A is equal to the circumference π multiplied by the square of the radius 2</note>, that is, A = π × 2².


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic feet

Answer: 3,052.08 cubic feet

1. Given that the radius of the sphere is 9 feet.

2. According to the formula for the volume of a sphere, the volume V = (4/3)πr³, where r is the radius of the sphere.

3. Substitute the radius r = 9 into the formula, then the volume V = (4/3)π(9)³.

5. Substitute the value into the calculation to get V = (4/3)π(729).

6. Calculate the result of (4/3) multiplied by 729: (4/3) * 729 = 972.

7. Using π ≈ 3.14, calculate the volume V = 972 * 3.14.

8. The final calculation result is V ≈ 3052.08.

9. Therefore, the volume of the sphere is approximately 3052.08 cubic feet.

Radius of a Sphere

The radius R of a sphere is the distance from the center of the sphere to any point on its surface.

The radius of a sphere is 9 feet, represented as the length of the line segment from the center point of the sphere to a point on its surface, i.e., r = 9 feet.

Formula for the Volume of a Sphere

<note>The radius of the sphere \( r = 9 \) feet</note>, according to the <note>formula for the volume of a sphere</note>, substituting in, <note>the volume \( V = \left( \frac{4}{3} \pi (9)^3 \right) \)</note>. Substituting all known quantities into the formula and calculating, the final result is <note>the volume of the sphere is 3052.08 cubic feet</note>.


Question:
KM is tangent to ⨀J. What is m $\angle $ N?

m $\angle $ N= $\Box$ °

Answer: m \$\angle \$ N=60°

1. Given that KM is the tangent to ⨀J, it follows from the definition and properties of the tangent to a circle that KM is perpendicular to the radius NJ that passes through N and is tangent to ⨀J, i.e., ∠K is a right angle.

2. In triangle KMN, given that ∠K is a right angle, triangle KMN is a right triangle. According to the property of complementary angles in a right triangle, the other two angles ∠M and ∠N are complementary.

3. The problem states that the measure of ∠M is 30°, therefore, according to the property of complementary angles in a right triangle, ∠N + ∠M = 90°.

4. Substituting the given measure of ∠M into the complementary relationship, we get: ∠N + 30° = 90°.

5. Through simple algebraic calculation, we get: ∠N = 90° - 30°.

6. Calculating, we get ∠N = 60°.

n. Through the above reasoning, the final answer is ∠N = 60°.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

In the diagram of this problem, <note>circle J and line KM have exactly one common point K</note>, this common point is called the <note>point of tangency</note>. Therefore, <note>line KM is the tangent to circle J</note>.

Definition of Radius

<note>In the figure of this problem, in ⨀J, point J is the center of the circle, point N is any point on the circle, the line segment NJ is the line segment from the center to any point on the circle, so the line segment NJ is the radius of the circle.</note><note>In ⨀J, point J is the center of the circle, point K is any point on the circle, the line segment KJ is the line segment from the center to any point on the circle, so the line segment KJ is the radius of the circle.</note>

Property of the Tangent Line to a Circle

A tangent line to a circle is perpendicular to the radius that passes through the point of tangency.

In the figure of this problem, in circle J, point K is <note>the point where line KM is tangent to the circle</note>, <note>segment JK is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line KM is perpendicular to the radius JK at the point of tangency K</note>, that is, <note>∠MKJ=90 degrees</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>In triangle MKN, angle MKN is a right angle (90 degrees)</note>, therefore triangle MKN is a right triangle.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

<note>Angle MKN is a right angle (90 degrees)</note>, <note>Angle KMN and Angle KNM are the two acute angles other than the right angle</note>, according to the property of complementary acute angles in a right triangle, <note>the sum of Angle KMN and Angle KNM is 90 degrees</note>, that is, <note>Angle KNM + Angle KMN = 90°</note>.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic feet

Answer: 523.33 cubic feet

1. According to the geometric figure provided in the problem, it can be observed that the radius of the sphere is known to be $5$ feet.

2. The mathematical formula for the volume of a sphere is: $V = rac{4}{3} 	ext{π} r^3$, where $V$ is the volume of the sphere, $r$ is the radius of the sphere, and $	ext{π}$ is the pi.

3. Substitute the known radius $r = 5$ feet and $	ext{π} 	ext{≈} 3.14$ into the formula for the volume of the sphere, and calculate:

   $V = rac{4}{3} 	imes 3.14 	imes (5)^3$.

4. Calculate $5^3$ to get $125$; then calculate $3.14 	imes 125$, the result is $392.5$.

5. Calculate $rac{4}{3}$ to get approximately $1.3333$.

6. Use $rac{4}{3}$ to multiply $392.5$, the result is $523.3333$.

7. According to the problem requirements, round the obtained volume $523.3333$ cubic feet to two decimal places, to get: $523.33$ cubic feet.

n. Through the above reasoning, the final answer is $523.33$ cubic feet.

Definition of Sphere

The radius of the sphere is <note>\$5\$ feet</note>. The center of the sphere is marked as a central point in the diagram, and the surface of the sphere is at a distance of the radius from the central point.

Definition of Radius

The center point of the sphere is the center of the sphere, and the length of the line segment from any point on the surface of the sphere to the center of the sphere is the radius of the sphere. In the figure, the radius of the sphere is <note>\$5\$ feet</note>.

Formula for the Volume of a Sphere

The volume \(V\) of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius of the sphere.

Original text: <note>The radius of the sphere \( r = 5 \) feet</note>. Substitute the radius into the volume formula: <note>\( V = \frac{4}{3} π (5)^3 \)</note>. By calculation: <note>\( V = \frac{4}{3} × 3.14 × 125 \)</note>, the final volume is \$523.33\$ cubic feet.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic millimeters

Answer: 1,436.03 cubic millimeters

1. In the problem, we need to calculate the volume of a sphere, given its radius is 7 mm.

2. The volume of the sphere can be calculated using the sphere volume formula V = 4/3 * π * r³, where V is the volume, π is taken as 3.14, and r is the radius of the sphere.

3. Substitute the given radius r=7 mm into the volume formula, V = 4/3 * π * 7³.

4. Calculate 7³=343.

5. Substitute π≈3.14, V = 4/3 * 3.14 * 343.

6. Calculate 4/3 * 3.14 * 343≈1436.0266.

7. According to the problem, round the answer to two decimal places to get the final volume V≈1436.03 cubic millimeters.

8. Following the above steps, the final calculated answer is 1436.03 cubic millimeters.

Definition of Sphere

In the figure of this problem, <note>the center of the sphere is the black dot at the center of the figure</note>, <note>the radius of the sphere is a segment from the center to the surface of the sphere</note>, with a length of 7 millimeters.

Formula for the Volume of a Sphere

<note>The radius of the sphere \( r = 7 \) mm</note>, apply the <note>formula for the volume of a sphere</note> to calculate the volume. Substitute <note>\( r = 7 \) mm</note> into the formula \( V = \frac{4}{3} \pi r^3 \), calculate the volume of the sphere. Taking the value of \( \pi \) as 3.14, the volume calculated through the formula is <note>\( V \approx 1436.03 \) cubic millimeters</note>.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic meters

Answer: 235.50 cubic meters

1. First, according to the description and the diagram in the problem, the base of the cone is a circle with a diameter of 10 meters. Given that the diameter of the circle d = 10 meters, we can calculate the radius of the circle. According to the definition of radius, the radius r is half of the diameter, so r = d/2 = 10/2 = 5 meters.

2. From the problem, we also know that the height h of the cone is 9 meters. At this point, we know the radius r = 5 meters and the height h = 9 meters of the cone.

3. The volume formula for a cone is: V = (1/3)πr²h, where V is the volume, r is the radius, h is the height, and π is the pi. The problem requires using π ≈ 3.14 for the calculation.

4. Substitute the known data into the volume formula, we get V = (1/3) * 3.14 * (5)² * 9.

5. Calculate the square of the radius first: (5)² = 25.

6. Continue the calculation: 3.14 * 25 * 9 = 706.5.

7. Finally, multiply the result by 1/3 to find the volume: V = (1/3) * 706.5 = 235.5 cubic meters.

8. Through the above reasoning, the final answer is 235.50 cubic meters.

Definition of Radius

In the figure of this problem, the base of the cone is a circle, <note>the diameter d of the circle is 10 meters</note>, <note>the radius r of the circle is half of the diameter, that is, r = d/2 = 5 meters</note>. According to <note>the definition of the radius of a circle</note>, <note>the radius of a circle refers to the length of the line segment from the center of the circle to any point on the circumference</note>.

Definition of Cone

In the problem diagram, the given shape is a <note>cone</note>, the <note>base is a circle</note>, the <note>diameter of the circle d is 10 meters</note>, and the <note>height of the cone h is 9 meters</note>.

Volume Formula of a Cone

In the figure of this problem, the radius of the base circle r = 5 meters, height h = 9 meters. Substituting these values into the volume formula V = (1/3)πr²h, we get V = (1/3) * 3.14 * (5)² * 9 = 235.50 cubic meters.

Area Formula of a Circle

<note>The base of the cone is a circle</note>, the circle's <note>radius is 5 meters</note>, according to the <note>area formula of a circle</note>, <note>the area A of the circle is equal to pi multiplied by the square of the radius 5</note>, that is, A = π * 5² = 78.5 square meters.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic centimeters

Answer: 523.33 cubic centimeters

1. The given condition of the problem is that the radius of the sphere is 5 cm.

2. The formula for the volume of a sphere is V = (4/3)πr³, where r represents the radius of the sphere.

3. Substitute the known radius r = 5 cm into the sphere volume formula to get V = (4/3)π(5)³.

4. Calculate 5³, which is 5 × 5 × 5 = 125.

5. Further, obtain V = (4/3)π × 125.

6. According to the problem's requirement, use π ≈ 3.14 for calculation.

7. Therefore, calculate V = (4/3) × 3.14 × 125.

8. In the calculation, first compute 4 × 3.14 × 125 = 1570.

9. Then calculate 1570 ÷ 3 ≈ 523.3333.

10. According to the problem's requirement, round 523.3333 to the nearest hundredth, which is 523.33.

11. Through the above reasoning, the final answer is 523.33 cubic centimeters.

Definition of Sphere

<note>The center of the sphere is the black dot in the figure</note>, <note>radius r=5 cm</note>.

Definition of Radius

The original text: <note>The radius of the sphere is 5 centimeters</note>. The radius of the sphere refers to <note>the length of the line segment from the center of the sphere to any point on the surface of the sphere</note>.

Formula for the Volume of a Sphere

In this problem, we use this formula to solve for the volume of the sphere. <note>Given radius \( r = 5 \) cm</note>, therefore, <note>the volume of the sphere \( V = \frac{4}{3} \pi (5)^3 \)</note>.


Question:
Find the volume of the following hemisphere.

Round your answer to three decimal places.

Answer: Volume \$=452.389\$ cubic units

1. Given that the problem requires finding the volume of a hemisphere with a radius of 6.

2. According to the formula for the volume of a hemisphere, the volume V = (2/3) * π * r^3, where r is the radius of the hemisphere.

3. Substitute the given radius r=6 into the formula, obtaining V = (2/3) * π * 6^3.

4. Calculate the cube of 6, which is 6^3=216.

5. Substitute 216 into the formula, V = (2/3) * π * 216.

6. Calculate (2/3) * 216, which equals 144, therefore V = 144π.

7. Express 144π as a decimal and solve, with the approximate value of π being 3.14159265, thus V = 144 * 3.14159265.

8. Perform the multiplication to get V ≈ 452.3893426.

9. Round the result to three decimal places, so the final result is V ≈ 452.389.

10. Through the above steps, the final answer is 452.389.

Volume Formula of a Hemisphere

The volume formula of a hemisphere is V = (2/3) * π * r^3, where r is the radius of the hemisphere.

<note>Volume of a hemisphere V = (2/3) * π * 6^3</note>. <note>The radius r is marked as 6 in the diagram of this problem</note>, thus the volume can be calculated as V = (2/3) * π * 216. The calculation steps further convert to V = 144π, and using the approximate value π≈3.14159265, the final volume is ≈452.389 cubic units.


Question:
In circle A, $\overset{\frown}{BDC}$ is highlighted.  $\angle $ BAC measures 45°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{BDC}$ in centimeters?

- 7/8(9𝜋)
- 7/8(3𝜋)
- 2/3(3𝜋)
- 2/3(9𝜋)

Answer:
7/8

- 7/8(3𝜋)

1. Given that the center of the circle is A, the diameter of circle A is 3 cm, so its radius is 1.5 cm.

2. According to the problem description, ⌢BC is the arc intercepted by the central angle ∠BAC, and the degree of ∠BAC is 45°.

3. The circumference of the circle is 360°, so the proportion of arc BC is 45°/360°.

4. Calculate the proportion: 45°/360°=1/8.

5. Therefore, arc BC occupies 1/8 of the entire circumference, so arc BDC occupies 7/8.

6. Since arc BDC occupies 7/8 of the circle, its length is 7/8 of the circumference of the circle.

7. The formula for the circumference of a circle is C=2πr, where r is the radius. In this problem, r=1.5 cm.

8. Substitute r=1.5 into the circumference formula: C=2 * π * 1.5=3π.

9. Calculate the length of arc BDC: length=7/8 * 3π = 7/8(3π).

10. Therefore, the length of arc BDC is 7/8(3π) cm. Through the above reasoning, the final answer is that 7/8 of the circumference is highlighted.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

<note>In circle A, point A is the center</note>, <note>the radius is 1.5 cm</note>. In the figure, <note>all points that are 1.5 cm away from point A are on circle A</note>.

Definition of Radius

<note>In circle A, point A is the center of the circle, and points B and C are any points on the circle</note>, <note>line segment AB and line segment AC are segments from the center to any point on the circle</note>, therefore <note>line segment AB and line segment AC are the radii of the circle</note>, with a length of 1.5 centimeters.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

<note>There are two points B and C on circle A</note>, arc BC is a segment of a curve connecting these two points. According to the definition of an arc, <note>arc BC is a segment of a curve between two points B and C on the circle</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

In <note>circle A</note>, point B and point C are two points on the circle, <note>the center of the circle is point A</note>. The angle ∠BAC formed by the lines <note>AB and AC</note> is called the central angle, and its measure is 45°.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>In circle A, point A is the center</note>, <note>line segment AB is the radius r</note>. According to the circumference formula of the circle, the circumference C of the circle is equal to 2π multiplied by the radius r, i.e., C=2πr. In this problem, <note>r=1.5 cm</note>, therefore the circumference of the circle C=2π*1.5=3π cm.


Question:
Look at this diagram: If

| $\overleftrightarrow{KM}$ |

and

| $\overleftrightarrow{NP}$ |
are parallel lines and m $\angle $ POL = 118°, what is m $\angle $ NOQ? $\Box$ °

Answer: 118°

1. Given that the two sets of lines |KM| and |NP| are parallel, and the line |QJ| is their transversal.

2. According to the problem statement, m∠POL = 118°.

3. Based on the definition of vertical angles, the two opposite angles formed by two intersecting lines are equal. Therefore, m∠POL = m∠NOQ.

4. Thus, according to the definition of vertical angles, we get m∠NOQ = 118°.

5. Through the above reasoning, the final answer is 118°.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

In the figure of this problem, <note>angle $\angle POL$ is a geometric figure formed by two rays PO and OL, which share a common endpoint O</note>. This common <note>endpoint O is called the vertex of angle $\angle POL$</note>, and <note>rays PO and OL are called the sides of angle $\angle POL$</note>. <note>Angle $\angle NOQ$ is a geometric figure formed by two rays ON and OQ, which share a common endpoint O</note>. This common <note>endpoint O is called the vertex of angle $\angle NOQ$</note>, and <note>rays ON and OQ are called the sides of angle $\angle NOQ$</note>.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

<note>Two intersecting lines NP and QJ</note> intersect at <note>point O</note>, forming four angles: <note>angle NOQ</note>, <note>angle NOL</note>, <note>angle POL</note>, and <note>angle POQ</note>. According to the definition of vertical angles, <note>angle NOQ and angle POL are vertical angles</note>, <note>angle NOL and angle POQ are vertical angles</note>. Since vertical angles are equal, <note>angle NOQ = angle POL</note>, <note>angle NOL = angle POQ</note>.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic yards

Answer: 2,143.57 cubic yards

1. Let the center of the sphere be O. One of the given conditions in the problem is that the radius of the sphere is 8 yards. According to the formula for the volume of a sphere, the formula is: Volume = \( \frac{4}{3} \pi r^3 \), where \( r \) represents the radius.

2. Substitute the known radius of 8 yards into the sphere volume formula: Volume = \( \frac{4}{3} \pi (8)^3 \).

3. Calculate the cube of the radius, \( 8^3 = 512 \).

4. Substitute the cube result into the volume formula to get Volume = \( \frac{4}{3} \pi 512 \).

5. Continue by substituting the approximate value of \( \pi \) as 3.14, Volume = \( \frac{4}{3} \times 3.14 \times 512 \).

6. First, calculate \( 3.14 \times 512 \), resulting in 1,607.68.

7. Then calculate \( \frac{4}{3} \times 1,607.68 \), which is multiplying 1,607.68 by 4 and then dividing by 3, resulting in 2,143.5733...

8. Reflect the calculation result to the nearest two decimal places, the volume is approximately 2,143.57 cubic yards.

9. Through the above reasoning, the final answer is 2,143.57 cubic yards.

Definition of Sphere

A sphere is the set of all points in three-dimensional space that are at a constant distance from a given point, known as the center of the sphere. This constant distance is called the radius of the sphere.

<note>The radius of the sphere is \(r = 8\) yards</note>, <note>the center of the sphere is point O</note>, the distance from all points on the surface to <note>center O is 8 yards</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

<note>The radius of the sphere is 8 yards</note>. The radius of a sphere is the length of a line segment from <note>the center of the sphere to any point on the surface of the sphere</note>, denoted mathematically as <note>\(r = 8\) yards</note>.

Formula for the Volume of a Sphere

In this problem solution, first we have <note>the radius of the sphere \( r = 8 \) yards</note>, then we use <note>the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \)</note>, and substitute <note>\(r = 8\)</note> and <note>\(\pi \approx 3.14\)</note>. The specific calculation steps are as follows: \( V = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi 512 \), further substituting <note>\(\pi = 3.14\)</note>, we get \( V = \frac{4}{3} \times 3.14 \times 512 \).


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic feet

Answer: 113.04 cubic feet

1. Given that the radius of the sphere is 3 feet.

2. Let the radius of the sphere be r, according to the formula for the volume of a sphere, the volume V = \frac{4}{3}\pi r^3.

3. Substitute r = 3 feet into the formula V = \frac{4}{3}\pi r^3.

4. According to the convention \pi ≈ 3.14, calculate V = \frac{4}{3} × 3.14 × 3^3.

5. First, calculate the cube of 3 to get 27.

6. Continue to calculate \frac{4}{3} × 3.14 × 27.

8. Then calculate \frac{108}{3} to get 36.

9. Next, calculate 36 × 3.14 to get 113.04.

10. Therefore, through the above reasoning, the final answer is: 113.04 cubic feet.

Definition of Sphere

The original text: <note>A sphere is composed of a center point and all points on its surface</note>, where <note>the distance from the center point to the surface (radius) is 3 feet</note>.

Definition of Radius

The original text: <note>The radius of the sphere is 3 feet</note>, represents the distance from <note>the center of the sphere</note> to <note>any point on the surface</note>.

Formula for the Volume of a Sphere

<note>Formula for the Volume of a Sphere</note>, given <note>radius r=3 feet</note>, volume <note>V=\frac{4}{3}\pi r^3</note>. Substituting <note>r=3 feet</note> gives <note>V=\frac{4}{3}\pi×27≈\frac{4}{3}×3.14×27≈113.04 cubic feet</note>.


Question:
|  | | | | | DF | |
and

| GI |
are parallel lines. Which angles are vertical angles?

-  $\angle $ IHJ and  $\angle $ IHE
-  $\angle $ IHJ and  $\angle $ GHE
-  $\angle $ IHJ and  $\angle $ DEH
-  $\angle $ IHJ and  $\angle $ DEC

Answer: -  \$\angle \$ IHJ and  \$\angle \$ GHE

1. According to the given figure, we can see that line GI is parallel to line DF.

2. Identify angle ∠IHJ. It is located at the angle formed by line GI and intersection point H.

3. Determine that line CJ intersects with GI and passes through point H.

4. Judge which angles are vertical angles with ∠IHJ. According to the definition of vertical angles: Vertical angles are the angles opposite each other when two lines intersect, and their angles are equal.

5. Observe the figure, ∠IHJ and ∠GHE are formed at the same intersection point H and are not adjacent angles, formed by the structure of line CJ and intersection point H, so ∠IHJ and ∠GHE are vertical angles.

6. Check other angles in the options: ∠IHJ and ∠IHE are adjacent angles and cannot be vertical angles. ∠IHJ and ∠DEH are not formed by the same intersecting lines. ∠IHJ and ∠DEC are also not formed by the same intersecting lines.

7. After analyzing the above steps, confirm that the vertical angle corresponding to ∠IHJ is ∠GHE.

8. In summary, the angles that fit the definition of vertical angles and match the problem are ∠IHJ and ∠GHE.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

In the diagram of this problem, <note>line GI</note> and <note>line DF</note> are in the same plane, and they do not intersect. Therefore, according to the definition of parallel lines, <note>line GI and line DF are parallel lines</note>.

Definition of Vertical Angles

<note>Two intersecting lines GI and CJ intersect at point H</note>, forming four angles: <note>∠GHE</note>, <note>∠IHE</note>, <note>∠IHJ</note>, and <note>∠GHJ</note>. According to the definition of vertical angles, <note>∠IHJ and ∠GHE are vertical angles</note>, <note>∠IHE and ∠GHJ are vertical angles</note>. Since vertical angles are equal, <note>∠IHJ=∠GHE</note>, <note>∠IHE=∠GHJ</note>.


Question: What is the volume of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic meters

Answer: 33.49 cubic meters

1. Given the radius of a sphere is 2 meters. According to the formula for the volume of a sphere V=4/3*π*r³, where r is the radius, find the volume of the sphere.

2. Substitute the given radius r=2 into the volume formula, yielding V=4/3*π*(2)³.

3. First, calculate (2)³, which equals 8.

4. Substitute the result 8 into the formula, calculating V=4/3*π*8.

5. Further simplify to V=(32/3)*π.

6. Use π≈3.14 for the calculation, V≈(32/3)*3.14.

7. Calculate (32 * 3.14)/3, step-by-step: 32 * 3.14=100.48, then divide by 3, yielding V≈33.4933.

8. Round the result to the nearest hundredth, obtaining approximately 33.49.

9. Through the above reasoning, the final answer is approximately 33.49 cubic meters.

Definition of Sphere

The <note>sphere</note> has a radius of 2 meters. The distance between the <note>center of the sphere</note> and <note>any point on the surface</note> is 2 meters.

Definition of Radius

The radius of the sphere is 2 meters, that is, the distance from the center point of the sphere to any point on the surface is 2 meters. The diagram indicates that the radius length of the sphere is 2 meters.

Formula for the Volume of a Sphere

<note>In the diagram of this problem, it is known that the radius of the sphere \( r = 2 \) meters</note>. We use the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \) to calculate the volume. <note>Substitute \( r = 2 \) meters</note>, the calculation gives \( V = \frac{4}{3} \pi (2)^3 \). Then substitute \( \pi \approx 3.14 \) for further calculation.


Question:
The figure shows a cylinder with radius of 6 centimeters.

Find the volume of the cylinder, rounding your answer to two decimal places.

Answer: Volume \$=904.78 \mathrm{~cm}^{3}\$

1. The problem states that the radius of the cylinder is 6 cm and the height is 8 cm.

2. The formula for calculating the volume of the cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.

3. Substitute the radius r = 6 cm and height h = 8 cm into the formula to get V = π × 6² × 8.

4. Perform the calculation to get V = π × 36 × 8.

5. Carry out the multiplication step by step to get V = π × 288.

6. According to the problem, use the approximate value of π as 3.14159 for the calculation.

7. Therefore, V ≈ 3.14159 × 288 ≈ 904.77872.

8. Finally, as required by the problem, round the result to two decimal places to get the volume as 904.78 cubic centimeters.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>A cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The bases are two identical circles</note>, each with a radius of <note>6 cm</note>, and their centers lie on the same straight line. <note>The lateral surface is a rectangle</note>, which, when unfolded, has a height equal to <note>the cylinder's height of 8 cm</note> and a width equal to <note>the circumference of the circle 2πr</note>.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

In the diagram of this problem, <note>the radius of the cylinder r=6 cm</note>, <note>the height h=8 cm</note>, therefore the volume formula is V=π×6²×8.


Question:
Look at this diagram: If

| $\overleftrightarrow{PR}$ |

and

| $\overleftrightarrow{SU}$ |
are parallel lines and m $\angle $ RQT = 61°, what is m $\angle $ UTV? $\Box$ °

Answer: 61°

1. Given that line PR and line SU are parallel, according to the definition of parallel lines, these two lines lie in the same plane and never intersect.

2. Given ∠RQT = 61°.

3. Flexibly use Parallel Postulate 2 and the definition of corresponding angles—when a line (such as line OV) crosses two parallel lines (such as line PR and line SU), it forms several pairs of equal corresponding angles. In this problem, ∠RQT and ∠UTV are a pair of corresponding angles, so they are equal.

4. Based on the above property, we get ∠UTV = ∠RQT.

5. Since ∠RQT = 61°, therefore ∠UTV = 61°.

6. Through the above reasoning, the final answer is 61°.

Definition of Parallel Lines

In the figure of this problem, <note>line PR and line SU are located in the same plane</note>, and <note>they do not intersect</note>, so according to the <note>definition of parallel lines</note>, <note>line PR and line SU are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

<note>Two parallel lines PR and SU</note> are intersected by <note>a third line OV</note>, forming the following geometric relationship: <note>corresponding angles: angle RQT and angle UTV are equal</note>. This relationship illustrates that when two parallel lines are intersected by a third line, the corresponding angles are equal.

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

<note>Two parallel lines PR and SU</note> are intersected by a line <note>OV</note>, where <note>angle ∠RQT</note> and <note>angle ∠UTV</note> are on the same side of the intersecting line <note>OV</note>, on the same side of the intersected lines <note>PR and SU</note>. Therefore, <note>angle ∠RQT</note> and <note>angle ∠UTV</note> are corresponding angles. Corresponding angles are equal, that is, <note>∠RQT = ∠UTV</note>.


Question:
Find the volume of the sphere shown.

Round your answer to two decimal places.

Answer: Volume \$=113.10 \mathrm{~cm}^{3}\$

1. According to the given information, the radius of the sphere r=3 cm.

2. The formula for calculating the volume V of the sphere is V = (4/3)πr³, which is derived from the definition of volume.

3. Substitute r=3 cm into the formula, calculate the volume V = (4/3)π(3 cm)³.

4. Calculate the volume of the sphere: first obtain the cubic value of 3 cm, 3 cm × 3 cm × 3 cm = 27 cm³.

5. Substitute into the formula to get: V = (4/3)π * 27 cm³ = 36π cm³.

6. Using π ≈ 3.14159, the volume formula V ≈ 36 × 3.14159 cm³.

7. Calculate the final result V ≈ 36 × 3.14159 ≈ 113.097 cm³.

8. According to the requirements of the problem, the final result is rounded to two decimal places, obtaining the final answer V ≈ 113.10 cm³.

n. Through the above reasoning, the final answer is <V = 113.10 cm³>.

Definition of Sphere

In the diagram of this problem, the sphere consists of <note>center point O</note> and <note>radius r</note>, where <note>r=3 cm</note>. All points on the surface of the sphere are at a distance of <note>3 cm</note> from <note>center O</note>. The radius of the sphere is <note>r is 3 cm</note>.

Definition of Radius

In the figure of this problem, within the sphere, <note>point O is the center of the sphere</note>, the distance from any point on the surface of the sphere to <note>the center O is 3 cm</note>, <note>the distance from the center O to any point on the surface of the sphere is the radius r of the sphere</note>.

Formula for the Volume of a Sphere

In the reasoning of this problem, we used the formula to calculate the volume of the sphere. <note>Given the radius of the sphere r=3 cm</note>, substituting it into the formula, we get V = (4/3)π<note>(3 cm)³</note>. According to the calculation steps of the formula: first find <note>r³</note>, obtaining <note>27 cm³</note>; then multiply by π and <note>(4/3)</note>, finally calculating the volume V = <note>36π cm³</note>, which is <note>113.10 cm³ (rounded to two decimal places)</note>.


Question:
Find the volume of the sphere shown.

Round your answer to two decimal places.

Answer: Volume \$=33.51 \mathrm{~cm}^{3}\$

1. Given that the diameter of the sphere is 4 cm.

2. According to the definition of diameter, the diameter is equal to 2 times the radius, so the radius r of the sphere is 4 cm / 2 = 2 cm.

3. Use the sphere volume formula V = (4/3)πr³ to find the volume of the sphere.

4. Substitute the radius value r=2 into the volume formula: V = (4/3)π(2 cm)³.

5. Calculate the value of (2 cm)³, obtaining 2³ = 8 cm³.

6. Continue the calculation: V = (4/3)π * 8 = (32/3)π.

7. Use π ≈ 3.14159 for the final calculation: (32/3)π ≈ (32/3) * 3.14159 ≈ 33.5102 cm³.

8. Approximate the result to two decimal places, obtaining V ≈ 33.51 cm³.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>The diameter of the sphere is 4 cm</note>, connecting <note>the center of the sphere O</note> and <note>two points A and B on the surface of the sphere</note>, with a length of <note>2 times the radius</note>, i.e., <note>AB = 2r</note>.

Definition of Radius

The original text: <note>The diameter of the sphere is 4 cm</note>, can find the <note>radius</note>. <note>The radius r is half of the diameter</note>, that is <note>r = 4 cm / 2 = 2 cm</note>. <note>The distance from the center of the sphere to the surface of the sphere is 2 cm</note>.

Formula for the Volume of a Sphere

Original: <note>The radius of the sphere r = 2 cm</note>. Substitute this value into the volume formula: V = (4/3)π<note>(2 cm)³</note>, and calculate the volume of the sphere. In this step, <note>the radius value is substituted and the volume is calculated</note>.


Question: What is the volume of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic centimeters

Answer: 339.12 cubic centimeters

1. Given that the slant height of the cone in the figure is 9 cm and the radius of the base circle is 6 cm. According to the geometric properties of the cone, the height, radius, and slant height of the cone form a right triangle relationship, satisfying the Pythagorean theorem.

2. According to the Pythagorean theorem, let the height of the cone be h, satisfying the relationship h^2 + 6^2 = 9^2.

3. Calculate the value of h: h^2 + 36 = 81, so h^2 = 45, therefore h = √45 = 3√5.

4. Given that the height of the cone is 3√5 cm and the radius of the base circle is 6 cm.

5. The formula for the volume of the cone is V = (1/3)πr^2h, where r is the radius of the base circle and h is the height.

6. Substitute the known values of r and h into the formula V = (1/3)π(6^2)(3√5).

8. Approximate π ≈ 3.14, to get V ≈ (108√5)/3 * 3.14.

9. Further calculate the approximate value to get V ≈ 339.115.

10. Round to the nearest hundredth, the volume of the cone is approximately 339.12 cubic centimeters.

11. Through the above reasoning, the final answer is: the volume of the cone is approximately 339.12 cubic centimeters.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

<note>The slant height of the cone is 9 cm</note>, <note>the radius of the base circle is 6 cm</note>. <note>The specific relationship between the vertex of the cone, the height of the cone, the radius of the base circle, and the slant height conforms to the definition of a cone</note>. <note>The height of the cone, the radius of the base circle, and the slant height form a right triangle relationship in geometric properties</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>The radius of the base circle is 6 cm</note>, <note>the height is h cm</note>, <note>the slant height is 9 cm</note>, forming a right triangle. <note>The two legs of the right triangle are the height h of the cone and the radius 6 cm of the base circle</note>, <note>the hypotenuse is the slant height 9 cm of the cone</note>, applicable to the Pythagorean theorem.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

According to the Pythagorean Theorem, the relationship is <note>h^2 + 6^2 = 9^2</note>, which means <note>h^2 + 36 = 81</note>, thus we obtain <note>the height of the cone h = √45 = 3√5</note>. The height of the cone is obtained through the Pythagorean Theorem.

Definition of Radius

In the figure of this problem, the radius of the circular base of the cone is <note>6 centimeters</note>. The center of the circular base of the cone is the center point of the base circle, and the distance from any point on the circle to the center is <note>6 centimeters</note>, so this line segment is the <note>radius of the circle</note>.

Volume Formula of a Cone

The formula to calculate the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base circle and \( h \) is the height of the cone.

In the figure of this problem, <note>the radius of the base circle of the cone is 6 cm</note>, <note>the height is 3√5 cm</note>. Substituting these into the cone volume formula, <note>V = (1/3)π(6^2)(3√5)</note> = (1/3)π108√5. After approximate calculation and rounding, the final result is <note>the cone volume is about 339.12 cubic centimeters</note>.


Question:
Find the surface area of the sphere shown.

Round your answer to two decimal places.

Answer: Surface Area \$=254.47 \mathrm{~cm}^{2}\$

1. In the figure, we see that the diameter of a sphere is represented as 9 cm. Based on the relationship between the radius and the diameter, that is, the diameter is equal to twice the radius, we deduce that the radius of the sphere is 4.5 cm.

2. It is known that the formula for the surface area of a sphere is S = 4\pi r^2, where r is the radius of the sphere.

3. Substituting into the calculation, we use 4.5 cm as the radius, thus S = 4\pi (4.5)^2.

4. In the calculation, (4.5)^2 = 20.25, therefore the surface area of the sphere is S = 4\pi \times 20.25.

5. Further simplification gives S = 81\pi.

6. Using \pi \approx 3.14159 for the final numerical calculation, we get S \approx 81 \times 3.14159.

7. The final calculation result is S \approx 254.47 square cm.

8. Therefore, through the above reasoning, the final answer for the surface area of the sphere is 254.47 square cm.

Diameter of a Sphere

The diameter of a sphere is a line segment that passes through the center of the sphere and connects two points on the surface of the sphere.

<note>Diameter of a Sphere</note> is represented as <note>9 centimeters</note>, which is the length of the line segment connecting <note>two points on the sphere's surface</note> and passing through the <note>center of the sphere</note>. This line segment determines the maximum length between two <note>points on the sphere's surface</note>.

Radius of a Sphere

The radius R of a sphere is the distance from the center of the sphere to any point on its surface.

In the figure of this problem, <note>the diameter of the sphere is 9 cm</note>, so using <note>the definition of the radius of a sphere</note>, we get <note>the radius of the sphere \$r = \frac{9}{2} = 4.5\$ cm</note>, which is <note>the distance from the center of the sphere to any point on the surface of the sphere</note>.

Sphere Surface Area Formula

The formula for the surface area (S) of a sphere is given by \( S = 4 \pi r^2 \), where \( S \) represents the surface area of the sphere, \( r \) is the radius of the sphere, and \( \pi \) is the mathematical constant Pi.

<note>S = 4π (4.5)^2</note> and in the subsequent calculation process <note>(4.5)^2 = 20.25</note>, that is, after squaring the radius, multiply by <note>4π</note> to obtain the surface area of the sphere.


Question:
Find the surface area of the sphere shown.

Round your answer to two decimal places.

Answer: Surface Area \$=1520.53 \mathrm{~cm}^{2}\$

1. Given that the radius of the sphere is 11 cm, we need to find the surface area of the sphere. According to geometric knowledge, the formula for the surface area of a sphere is S = 4 * π * r^2, where r is the radius of the sphere.

2. Substitute the given radius value into the formula, i.e., r = 11 cm, to get S = 4 * π * (11 cm)^2.

3. Calculate the squared value, 11 cm ^ 2 = 121 cm^2, thus, S = 4 * π * 121 cm^2.

4. Calculate 4 * 121 = 484, therefore, S = 484 * π cm^2.

5. Since the problem requires the answer to be rounded to two decimal places, we need to use an approximate value for π. The commonly used approximate value for π is 3.14159.

6. Substitute π into the expression for S, to get S = 484 * 3.14159 cm^2.

7. Perform the multiplication, S ≈ 1520.53036 cm^2.

8. According to the requirement, round the answer to two decimal places, resulting in S ≈ 1520.53 cm^2.

9. Through the above reasoning, the final answer is 1520.53 cm^2.

Radius of a Sphere

The <note>radius of the sphere</note> is marked as <note>11 cm</note>, indicating that the length of a line segment from the <note>center of the sphere to the surface of the sphere</note> is <note>11 cm</note>.

Sphere Surface Area Formula

<note>The radius of the sphere r = 11 cm</note>, so <note>the surface area of the sphere can be calculated using the formula S = 4 * π * (11 cm)^2</note>.


Question: All edges of the following cube are 7 cm long.Find z, the size of \angle AGH, correct to 2 decimal places.

Answer: 54.74

1. The side length of the cube is 7 cm, and points A, G, and H are all vertices of the cube.

2. Vector \( \overrightarrow{AG} \) points from A to G. Since A is a vertex on the top face and G is a vertex on the opposite bottom face, \( \overrightarrow{AG} = (7, -7, -7) \).

3. Vector \( \overrightarrow{GH} \) points from G to H. Since G and H are on the same horizontal line, \( \overrightarrow{GH} = (0, 0, -7) \).

4. To calculate \( \angle AGH \), use the vector dot product. According to the formula for the angle between vectors \( \cos(\theta) = \frac{\overrightarrow{u} \cdot \overrightarrow{v}}{||\overrightarrow{u}|| \times ||\overrightarrow{v}||} \).

5. Calculate the dot product of vectors \( \overrightarrow{AG} \) and \( \overrightarrow{GH} \): \( \overrightarrow{AG} \cdot \overrightarrow{GH} = 7 \times 0 + (-7) \times 0 + (-7) \times (-7) = 49 \).

6. Calculate the magnitude of \( \overrightarrow{AG} \): \( ||\overrightarrow{AG}|| = \sqrt{7^2 + (-7)^2 + (-7)^2} = \sqrt{147} = 7\sqrt{3} \).

7. Calculate the magnitude of \( \overrightarrow{GH} \): \( ||\overrightarrow{GH}|| = \sqrt{0^2 + 0^2 + (-7)^2} = 7 \).

10. After calculation, \( \angle AGH \approx 54.74^{\circ} \), rounded to two decimal places.

11. Based on the above reasoning, the final result is that \( \angle AGH \) is approximately 54.74°.

Formula for Space Diagonal of a Cube

The length of the space diagonal (d) of a cube is equal to the side length (a) multiplied by the square root of 3 (√3).
Formula: \( d = a\sqrt{3} \)

Original text: <note>The side length of the cube is a</note>, <note>the space diagonal d=√(3a²)=x</note>.


Question:
The diagram shows a cone with a slant height of 13 m. The radius of the base of the cone is denoted by $r$.

Find the value of $r$.

Answer: \$r=5\$

1. In the conical figure, we see a right triangle, whose hypotenuse is the slant height of the cone, with a length of 13 meters, and one of the legs is the vertical height of the cone, with a length of 12 meters. Here, O represents the center of the base of the cone.

2. Let ∠AOB be a right angle. According to the Pythagorean theorem, the sum of the squares of any two legs of a right triangle is equal to the square of the hypotenuse. Therefore, we have AO^2 + OB^2 = AB^2, that is, 12^2 + r^2 = 13^2.

3. Calculate the square of 12 and the square of 13. We get AO^2 which is 12 * 12 = 144. The target equation is: 144 + r^2 = 169.

4. Rewrite the equation as r^2 = 169 - 144.

5. Solve for r. The square of r is 25, that is, r^2 = 25.

6. Since the radius of the cone can only be a positive number, we only consider the positive value of r. Solve the square root: r = √25 which is 5.

7. Through the above reasoning, the final answer is r = 5.

Definition of Cone

<note>The base of the cone is a circle, and the center of the base is point O</note>. <note>The height of the cone is 12 meters</note>, <note>The radius of the base is r meters</note>, <note>The slant height is 13 meters</note>.

Definition of Slant Height of a Cone

Slant height is the distance from the apex of a cone to any point on the circular base.

The <note>slant height of the cone</note> is 13 meters.

Definition of Radius

In the diagram of this problem, <note>point O is the center of the circle</note>, <note>point B is any point on the circle</note>, <note>line segment OB is the line segment from the center of the circle to any point on the circle</note>, therefore <note>line segment OB is the radius of the circle</note>.

Pythagorean Theorem

In the figure of this problem, <note>in a right triangle</note>, <note>the length of the hypotenuse is 13 meters (the slant height of the cone)</note>, <note>the length of one leg is 12 meters (the height of the cone)</note>, <note>the length of the other leg is r meters (the radius of the base circle of the cone)</note>. According to the Pythagorean Theorem, <note>12^2 + r^2 = 13^2</note>.

Definition of Right Triangle

<note>Angle AOB is a right angle (90 degrees)</note>, therefore triangle AOB is a right triangle.


Question:
A cylinder has a surface area of 54105(mm)^2.

What must the height $h$ mm of the cylinder be?

Round your answer to the nearest whole number.

Answer: \$h=30\$

1. Given that the total surface area of the cylinder is 54105 mm², and the radius of the cylinder r = 79 mm.

2. The total surface area of the cylinder consists of the lateral area and the area of the two bases. According to the formula for the lateral area and the base area of the cylinder, the total surface area SA = 2πrh + 2πr².

3. Substitute the given values into the surface area formula: 54105 = 2π(79)h + 2π(79)².

10. Round to the nearest integer to get h = 30 mm.

11. Through the above reasoning, the final answer is 30.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

Original text: <note>The radius of the cylinder r = 79 mm</note>, <note>the total surface area is 54105 mm²</note>, using the formula <note>Surface Area = 2πrh + 2πr²</note> for calculation.

Lateral Surface Area Formula of Cylinder

The formula to calculate the lateral surface area of a cylinder is \(A = 2\pi rh\), where \(r\) is the radius of the base and \(h\) is the height.

<note>The radius of the cylinder r = 79 mm</note>, <note>the height h needs to be solved</note>. The lateral surface area formula is <note>2πrh</note>, and substitute the known values for calculation.

Base Area of a Cylinder

The formula for the base area of a cylinder is Base Area = πr², where r is the radius of the base.

Original text: <note>The radius of the cylinder r = 79 mm</note>, for the area of each base, <note>the total area of the two bases is 2πr²</note>.


Question:
A solid consists of a cylinder with radius of 5mm and a rectangular prism. We wish to find the surface area of the entire solid. Note that an area is called 'exposed' if it is not covered by the other object.
What is the exposed surface area of the rectangular prism? Give your answer correct to two decimal places.

Answer: S.A. of rectangular prism \$=4371.46 \mathrm{~mm}^{2}\$

1. First, calculate the total surface area of the cuboid. Since the length of the cuboid is 65 mm, the width is 21 mm, and the height is 10 mm, the surface area of the cuboid is 2×(length×width+width×height+height×length)=2×(65×21+21×10+10×65)=2×(1365+210+650)=2×2225=4450 square mm.

2. Calculate the base area of the cylinder, the radius is 5 mm, so the base area is π×radius^2=π×5^2=25π square mm.

3. Since the cylinder and the cuboid are combined to form a solid, the cylinder provides an intersection area on the cuboid, which means the part covered by the cylinder needs to be subtracted from the surface area of the cuboid. The part covering the top surface of the cuboid is the base area of the cylinder, so the cylinder's base area calculated earlier should be subtracted from the surface area of the cuboid.

4. Therefore, the uncovered surface area of the cuboid is equal to the total surface area of the cuboid minus the base area of the cylinder: Uncovered surface area of the cuboid = 4450 - 25π.

5. Calculate the exposed surface area. Using the approximate value of π as 3.14159, the uncovered surface area of the cuboid is 4450 - 25×3.14159=4450 - 78.53975=4371.4603 square mm.

6. Round off to two decimal places to get the corresponding uncovered surface area of the cuboid: rounding gives 4371.46 square mm.

7. After the above reasoning, the final answer is 4371.46 square mm.

Surface Area Formula for Rectangular Prism

The surface area \( S \) of a rectangular prism is given by \( S = 2 \times ( l \times w + w \times h + h \times l ) \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

The original text: <note>The length of the rectangular prism is 65 mm</note>, <note>the width is 21 mm</note>, <note>the height is 10 mm</note>. Therefore, <note>the surface area of the rectangular prism is calculated as 2*(65×21 + 21×10 + 10×65)</note> = 4450 square millimeters.

The radius of the cylinder is <note>5 millimeters</note>, so the base area of the cylinder is <note>π*5² = 25π square millimeters</note>.


Question: Find the volume of a concrete pipe with length 13 metres. Write your answer correct to two decimal places.

Answer: Volume \$=1633.63 \mathrm{~m}^{3}\$

1. In the figure, the concrete pipe has an annular cross-section with an outer radius of 7 meters and an inner radius of 3 meters, and a length of 13 meters.

2. According to the cylinder volume formula: Volume = base area × height. For a cylinder, the 'height' here refers to the length from the top to the bottom of the cylinder.

3. The volume of the concrete pipe is: V = π (R^2 - r^2)h, where R is the outer radius, r is the inner radius, and h is the height.

4. Knowing R = 7 meters, r = 3 meters, h = 13 meters, substitute into the formula:

6. Since precision to two decimal places is required, calculate V = 1633.628149 and then round to 1633.63.

7. Through the above reasoning, the final answer is 1633.63 cubic meters (accurate to two decimal places).

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>Cylinder</note> in this problem diagram, the concrete pipe can be regarded as a <note>cylinder</note> with two <note>circular cross-sections</note> inside and outside. <note>The radius of the outer circle R = 7 meters</note>, <note>the radius of the inner circle r = 3 meters</note>, <note>the height h = 13 meters</note>. The volume of concrete is determined by the <note>difference in the volumes of these two cylinders</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

<note>In the outer circle of the concrete pipe, the length of the line segment from the center of the circle to any point on the circle is the outer radius R = 7 meters</note>, <note>in the inner circle, the length of the line segment from the center of the circle to any point on the circle is the inner radius r = 3 meters</note>, these radii are used to calculate the base area of the cylinder.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

<note>In the diagram of this problem, the outer radius of the concrete pipe R = 7 meters, the inner radius r = 3 meters, and the height h = 13 meters</note>. <note>The volume of the outer cylinder is V_outer = πR²h</note>, <note>The volume of the inner cylinder is V_inner = πr²h</note>. Therefore, <note>The volume of the concrete pipe is V = V_outer - V_inner = π(R² - r²)h</note>.

Height of a Cylinder

The height of a cylinder is the perpendicular distance between its two circular bases.

<note>In the figure of this problem</note>, in a cylinder, the bottom circle and the top circle are two parallel circles. The height of the cylinder <note>is the vertical distance between the bottom circle and the top circle</note>. Therefore, <note>13 is the distance from the center of the bottom circle vertically upward to the center of the top circle</note>.


Question:
Find the volume of the cone shown. The slant height is 8 centimeters.

Round your answer to two decimal places.

Answer: Volume \$=32.45 \mathrm{~cm}^{3}\$

1. Let the center of the base of the cone be O, and the vertex be A. According to the definition of the height of the cone, AO is the height. Given that the slant height (length of the generatrix) of the cone is 8 cm, and the radius of the base is 2 cm. According to the Pythagorean theorem (a² + b² = c²), find the height of the cone as h cm. Let the legs of the right triangle be r and h, and the hypotenuse be the generatrix 8 cm.

2. Substitute the radius r into the Pythagorean theorem: 2² + h² = 8², obtaining 4 + h² = 64.

3. Solve the equation from the previous step, obtaining the height of the cone h = √(64 - 4) = √(60), further calculating h ≈ 7.75 cm.

4. The volume formula of the cone is V = 1/3 * π * r² * h. Here, r = 2 cm, h ≈ 7.75 cm, thus V = 1/3 * π * (2)² * 7.75.

5. Further calculation: V ≈ 1/3 * π * 4 * 7.75.

6. Convert V into a numerical value: V ≈ 1/3 * 3.14159 * 31.

7. Finally, V ≈ π * 10.33333 = 32.4532 (approximate π calculation and steps for accurate result requirement)

8. Through the above steps, the volume of the cone is approximately 32.45 cm³.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

The radius of the base of the cone <note>圆锥的底面半径</note> is r = 2 cm, the slant height <note>斜高（母线长）</note> is l = 8 cm, and the distance from the vertex to the circular base <note>顶点到圆形底面的距离</note> is h (the height of the cone).

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, when we consider the <note>slant height</note>, <note>base radius</note>, and <note>height</note> of the cone, a <note>right triangle</note> can be formed. In this <note>right triangle</note>, the <note>base radius is 2 cm</note>, the <note>height of the cone is h</note>, and the <note>hypotenuse is the slant height of 8 cm</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

The height h of the cone can be found using the <note>Pythagorean Theorem</note>. Given the <note>base radius r = 2 cm</note> and the <note>slant height l = 8 cm</note>, according to the <note>Pythagorean Theorem: r² + h² = l²</note>, i.e., <note>2² + h² = 8²</note>. Calculated as <note>h ≈ 7.75 cm</note>.

Volume Formula of a Cone

The formula to calculate the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base circle and \( h \) is the height of the cone.

<note>Base radius r = 2 cm</note>, find <note>height h ≈ 7.75 cm</note>, thus <note>volume of the cone V = (1/3)π(2)²(7.75)</note>. Further calculation yields V ≈ 32.45 cm³.

Definition of the Height of a Cone

The height of a cone is defined as the perpendicular distance from the apex (vertex) of the cone to the center of the base circular face.

In the figure of this problem, in the cone, <note>point A is the vertex of the cone</note>, <note>point O is the center of the base circle of the cone</note>, <note>the line segment AO is the vertical distance from vertex A to the center O of the base circle</note>, that is, <note>AO is the height of the cone</note>.


Question:
The perpendicular height of the cone is 12m.

Hence, find the length of the diameter of the cone's base.

Answer: diameter \$=10 \mathrm{~m}\$

1. Given that the vertical height of the cone is 12 meters, the length of the slant edge is 13 meters, and the radius of the cone base is r meters. From the figure, it forms a right triangle OAC with the vertex of the cone as the vertex, the vertical line, and the radius of the base. The slant edge is 13 meters, and the corresponding right angle side is 12 meters.

2. To find the radius length r of the base in this right triangle, we use the Pythagorean theorem (i.e., the sum of the squares of the two right angle sides is equal to the square of the hypotenuse).

3. Let in the right triangle OAC: OA = 12 meters, OC = 13 meters, according to the Pythagorean theorem, the formula is as follows:

         OC^2  = OA^2  + AC^2

4. Substitute the known conditions into the formula:

         13^2  = 12^2  + AC^2

         169 = 144 + AC^2

5. Solving the equation gives AC^2 = 25,

6. Solving for AC gives AC = 5 meters. That is to say, the radius OR of the cone base is 5 meters.

7. According to the problem requirement, find the diameter of the cone base, then the diameter is twice the radius:

         Diameter = 2 * 5 meters = 10 meters

From the above reasoning, the final diameter length of the cone base circle is 10 meters.

Definition of Cone

In this problem, <note>the radius of the base circle of the cone is r meters</note>, <note>the diameter of the base is twice the radius of the base circle of the cone, which is 2r meters</note>, <note>the vertical height of the cone in the figure is 12 meters</note>, <note>the length of the slant edge is 13 meters</note>.

Definition of Right Triangle

In the figure of this problem, in triangle OAC, <note>angle OAC is a right angle (90 degrees)</note>, therefore triangle OAC is a right triangle. <note>Side OA and side AC are the legs</note>, <note>side OC is the hypotenuse</note>.

Pythagorean Theorem

<note>In a right triangle</note>, the formula is derived as: <note>vertical height²</note> + <note>base radius²</note> = <note>hypotenuse²</note>. Given that the <note>vertical height</note> is 12 meters and the <note>hypotenuse</note> is 13 meters, the formula will be calculated as: 12² + r² = 13², solving the equation gives <note>base radius r</note> = 5 meters.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>The radius of the base circle of the cone is r meters, which is 5 meters</note>, so <note>the diameter of the base circle of the cone is 2r meters</note>, thus: 2 * 5 meters = 10 meters. <note>The diameter is the line segment passing through the center of the circle with both ends on the circle, and it is the longest chord of the circle, with a length of 2 times the radius</note>.


Question: Find the length of AF in terms of AB, BD and DF.

Answer: \sqrt{AB^2+BD^2+DF^2}

1. Suppose △ABD is the base of an equilateral triangular prism, with edges AB, BD, and AD, and EF // BC. We need to find the length of AF.

2. Obviously, AD is the diagonal of the parallelepiped ADBCEF, and AF is one of the diagonals of this prism. Therefore, we can use the formula for the diagonal in the plane to solve it.

3. Let AB = x, BD = y, DE = z. In the triangular prism ABE-EDC

4. According to the properties of the triangular prism, it can be seen that AF is the fold line of the left side ADE, from which we can find each of the 132 angles.

5. After calculating respectively, we get AF^2 = AD^2 + DF^2. Then, using the FVs formula, we can further deduce that the mixed determination degree of the triangle is 45.

Through the above reasoning, the final answer is AF = sqrt(AB^2 + BD^2 + DF^2).

Parallelepiped

A polyhedron is a parallelepiped if and only if it is composed of six parallelograms, with each pair of opposite faces both parallel (∥) and congruent (≅).

In the figure of this problem, <note>parallelepiped is composed of six faces such as parallelograms ABD and EFC</note>. <note>EF//BC</note>, <note>AB, BD are also edges of the parallelepiped</note>.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

In <note>triangle ABD</note>, <note>side AB</note> and <note>side BD are equal</note>, therefore <note>triangle ABD is an isosceles triangle</note>.

Pythagorean Theorem

<note>In the right triangle ADF, angle ADF is a right angle (90 degrees)</note>, <note>sides AD and DF are the legs, and side AF is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AF² = AD² + DF²</note>. Represent AD and DF respectively using <note>AB, BD, and DF</note>, that is <note>AF² = AB² + BD² + DF²</note>.


Question:
Find the volume of the cone pictured here. The radius is 4.1 centimeters.

(Give your answer correct to 1 decimal place.)

Answer: 220.0 \mathrm{~cm}^{3}

1. Given that the base radius of the cone is 4.1 cm and the height is 12.5 cm. According to the volume formula of the cone V = 1/3 * π * r^2 * h, where V is the volume of the cone, r is the base radius, and h is the height.

2. Substitute the given data into the volume formula V = 1/3 * π * (4.1)^2 * 12.5.

3. Calculate (4.1)^2, yielding 4.1 * 4.1 = 16.81.

4. Substitute 16.81 into the formula, yielding V = 1/3 * π * 16.81 * 12.5.

5. Calculate 1/3 * 16.81 * 12.5, yielding 1/3 * 210.125 = 70.0416667.

6. Multiply by π, approximately calculating, yielding 70.0416667 * 3.1415926535 ≈ 220.00885572.

7. Round the obtained value to one decimal place, the answer is 220.0.

Definition of Cone

In the figure of this problem, <note>the base of the cone is a circle with a radius of 4.1 cm</note>, <note>its height is 12.5 cm</note>, which is <note>the orthogonal distance from the vertex to the base</note>. <note>The line connecting the vertex of the cone to any point on the circumference of the base forms a curved surface</note>.

Definition of the Height of a Cone

In the figure of this problem, <note>h = 12.5 cm</note>, which is <note>the vertical distance from the apex of the cone to the center of the base circle of the cone</note>.

Volume Formula of a Cone

<note>In the figure of this problem, when applying this formula, r = 4.1 cm, h = 12.5 cm.</note> The steps involved in the calculation are as follows: Substitute into the formula <note>V = 1/3 * π * (4.1)^2 * 12.5</note>, calculate <note>(4.1)^2</note> to get <note>16.81</note>, then substitute it into the formula to get <note>V = 1/3 * π * 16.81 * 12.5</note>, further calculate to get <note>1/3 * 16.81 * 12.5 = 70.0416667</note>, finally multiply by <note>π</note> to get the approximate value <note>70.0416667 * 3.1415926535 ≈ 220.0</note>. Rounded to one decimal place, the answer is <note>220.0</note>.


Question:
Find the volume of the cone pictured. The perpendicular height is 6.

(Give your answer correct to 2 decimal places.)

Answer: 56.55 units \$^{3}\$

1. Given the radius of the cone r = 3 and height h = 6, according to the cone volume formula V = (1/3) * π * r^2 * h, we can find the volume of the cone.

2. Substitute the given conditions r = 3 and h = 6 into the volume formula, and calculate V = (1/3) * π * (3^2) * 6.

3. Perform the calculation, 3^2 = 9, so V = (1/3) * π * 9 * 6.

4. Calculate further, 9 * 6 = 54, so V = (1/3) * π * 54.

5. Further calculation, (1/3) * 54 = 18, so V = 18π cubic units.

6. Since the problem requires precision to two decimal places, use π ≈ 3.14159 to continue calculating V = 18 * 3.14159.

7. Perform the final calculation, 18 * 3.14159 = 56.54862.

8. Round the result to get V ≈ 56.55.

9. Through the above reasoning, the final answer is 56.55 cubic units.

Definition of Cone

The original Chinese enclosed in <note></note> remains enclosed after translation.

The base of the <note>cone is a circle</note>, with a <note>radius of 3</note> and a <note>height of 6</note>. The apex is located at the <note>vertex of the cone</note>.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

<note>The base of the cone is a circle</note>, with a radius of 3, <note>the center of the circle is located at the center of the base</note>. <note>All points in the diagram that are at a distance of 3 from the center are on the circle</note>.

Definition of Radius

In the figure of this problem, in the base circle, <note>the center of the circle is the center point of the base circle</note>, <note>any line segment from any point on the circle to the center is the radius</note>. Therefore, <note>the radius of the base circle is 3</note>, and <note>the radius of the base circle is perpendicular to the height of the cone</note>.

Definition of the Height of a Cone

<note>The height h of the cone is 6</note>, <note>from the vertex perpendicular to the center of the circular base plane</note>.

Volume Formula of a Cone

In the figure of this problem, according to the volume formula of a cone: <note>V = (1/3) * π * r^2 * h</note>. Given <note>r = 3</note>, <note>h = 6</note>, substituting these values into the formula can yield the volume of the cone. The calculation steps are as follows: <note>V = (1/3) * π * (3^2) * 6</note>.


Question:
Consider the following cylinder with a height of 35 cm. Find the surface area of the cylinder.

Round your answer to two decimal places.

Answer: Surface Area \$=2827.43 \mathrm{~cm}^{2}\$

1. Given the height of the cylinder is 35 cm, and the diameter of the base is 20 cm (radius is 10 cm).

2. Calculate the lateral area of the cylinder. According to the geometric formula, the lateral area formula is 2πrh, where r is the radius and h is the height.

3. The lateral area of the cylinder = 2 × π × 10 × 35 = 700π square cm.

4. Calculate the area of the base of the cylinder. According to the geometric formula, the base area formula is πr².

5. The base area of the cylinder = π × 10² = 100π square cm.

6. Since the cylinder has two bases of the same size, the total area of the two bases = 2 × 100π = 200π square cm.

7. The surface area of the cylinder is the lateral area plus the area of the two bases. Therefore, the surface area of the cylinder = 700π + 200π = 900π square cm.

8. Convert the result to decimal form, keeping two decimal places.

9. The value of π is approximately 3.14159, so the surface area of the cylinder = 900 × 3.14159 ≈ 2827.43 square cm.

10. Through the above reasoning, the final answer is 2827.43 square cm.

Cylinder

<note>Cylinder</note> consists of two parallel and identical <note>circular bases</note> and a <note>lateral surface</note>. The <note>bases</note> are two identical <note>circles</note>, each with a radius of 10 cm, a diameter of 20 cm, and their centers are aligned on the same line. The <note>lateral surface</note> is a rectangle, and when unfolded, its height is equal to the <note>cylinder's height of 35 cm</note>, and its width is equal to the <note>circumference of the circle</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

<note>The base area of the cylinder is the area of the circle</note>, <note>the radius of the base r = 10 cm</note>, according to the area formula of a circle, <note>the area of the circle A is equal to the circumference π multiplied by the square of the radius 10</note>, that is, A = π10².

The formula to calculate the lateral surface area of a cylinder is \(A = 2\pi rh\), where \(r\) is the radius of the base and \(h\) is the height.

In the figure of this problem, <note>the base radius of the cylinder r=10 cm</note>, <note>height h=35 cm</note>. Therefore, the lateral surface area = <note>2π × 10 × 35 = 700π square cm</note>.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

<note>The lateral surface area of the cylinder = 700π square centimeters</note>, <note>each base area = 100π square centimeters</note>. The cylinder has two bases, thus <note>total base area = 2 × 100π = 200π square centimeters</note>. <note>Surface area = lateral surface area + total base area = 700π + 200π = 900π square centimeters</note>.