Question:
Consider the following hemisphere. Find the total surface area.

Round your answer to three decimal places.

Answer: Surface Area \$=603.186\$ units \$^{2}\$

1. Given that the figure is a hemisphere with a radius of 8, according to the definition of a hemisphere, it includes half the surface area of a sphere and one base area.

2. First, find the surface area S of the sphere using the formula S=4πr². Here, r is the radius of the hemisphere, which is r=8.

3. Substitute into the formula to calculate, from S=4π(8)² we get S=256π.

4. According to the problem, we need half of the sphere's surface area, calculated from 256π: 256π/2 gives half of the sphere's surface area as 128π.

5. Then, find the area of the base circle using the formula for the area of a circle A=πr², with r=8. Substitute into the formula A=π(8)² to calculate, we get A=64π.

6. Finally, calculate the total surface area of the hemisphere (hemisphere surface area + base area), which is 128π (part of the sphere's surface area) + 64π (base surface area), resulting in 192π.

7. After calculation, first replace π with the decimal standard, π is approximately 3.14159, 192π results in 603.18576 square units.

8. The problem requires retaining three decimal places, so we truncate the result to get 603.186.

9. Through the above reasoning, the final answer is 603.186.

Definition of Hemisphere

A hemisphere is a three-dimensional geometric shape that constitutes half of a sphere, including half of the sphere's surface area and a circular base area.

Original text: <note>A hemisphere consists of half of a sphere</note>. The surface area of a hemisphere includes <note>the hemispherical surface part and a base area</note>. The hemispherical surface part is <note>half of the sphere's surface area</note>, and the base area is <note>a circle, identical to the sphere's cross-section</note>. Therefore, the surface area of a hemisphere xxx includes the hemispherical surface part and the base area.

Definition of Radius

<note>The radius is 8 units</note>, which means the distance from <note>the center of the hemisphere to the surface of the sphere is 8 units</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>Sphere Surface Area Formula</note> is used to calculate the surface area of the entire sphere, <note>radius r=8</note>, substituting into the formula, the surface area of the entire sphere is <note>S = 4π(8)² = 256π</note>.

Area Formula of a Circle

In the figure of this problem, the base of the hemisphere is a circle, <note>the radius of the circle is 8</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 8, which is A = π8²</note>.


Question:
Calculate the volume of the solid. Assume that the solid is a quarter of a cylinder. The height is 15 cm.

Round your answer to one decimal place.

Answer: Volume \$=577.3 \mathrm{~cm}^{3}\$

1. Given that the base of the quarter-cylinder is a quarter circle with a radius of 7 cm, according to the cylinder volume formula, the volume of the cylinder V = base area A × height h.

2. The area formula of a circle is A = πr², where r is the radius of the circle. In this problem, r = 7 cm, so the area of the complete circle is π*(7)² = 49π square cm.

3. Since the base is a quarter of the circle, the area of the quarter part is one quarter * 49π = 12.25π square cm.

4. Given that the height h of the quarter-cylinder is 15 cm, applying the cylinder volume formula V = base area × height, we get V = 12.25π * 15.

5. Simplifying the expression for V, we get V = 183.75π cubic cm.

6. Approximating π as 3.14159, the volume is approximately V ≈ 183.75*3.14159 = 577.299 cubic cm.

7. Rounding the calculated result to one decimal place, we get approximately 577.3 cubic cm.

8. Finally, through the above reasoning steps, the volume of the quarter-cylinder is determined to be 577.3 cubic cm.

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>The base area A is the area of a quarter circle</note>, <note>the radius of the quarter circle r = 7 cm</note>, <note>height h = 15 cm</note>. According to the volume formula of the cylinder, the problem's <note>volume of the quarter cylinder equals the base area 12.25π square cm multiplied by the height 15 cm</note>, i.e., V = 12.25π × 15 = 183.75π cubic cm.

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 radius of the circle is 7 cm</note>, according to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to pi multiplied by the square of the radius 7</note>, that is, A = π × 7² = 49π square cm. Since <note>the base of a quarter circle is one-fourth of the circle</note>, therefore <note>the base area of the quarter circle is 49π ÷ 4 = 12.25π square cm</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: 763.02 cubic millimeters

1. According to the diagram in the problem, the base of the cone is a circle, and the line segment between the vertex and the center of the base is the height, which is the definition of the geometric shape of the cone.

2. It is known that the slant height of the cone is 9mm (measured from the vertex of the cone to any point on the circumference of the base).

3. Since the base of the cone is a circle, and the slant height of the cone is known to be 9mm, and the height of the cone and the base form the two legs of a right triangle, we can use the Pythagorean theorem to calculate the radius of the base circle.

4. In this right triangle, the hypotenuse is 9mm (i.e., the slant height of the cone), one of the legs is the height of the cone (also 9mm), and we use r to represent the radius of the base circle, thus we get the equation: 9^2 = r^2 + 9^2 (applying the Pythagorean theorem, a^2 + b^2 = c^2).

5. Simplifying the equation, we get: 81 = r^2 + 81.

6. By solving the equation, we get: r^2 = 81 - 81 = 0, which means r = 0. This is geometrically impossible, so we re-examine the information given in the problem.

7. In the diagram, there may be a misunderstanding; according to the diagram, the actual side line is not considered as the slant height; we need to assume at least half of the diameter of the base to determine the radius of the base of the cone.

8. Assuming the diameter of the cone is 9mm, then the radius of the base r = 9mm / 2 = 4.5mm.

9. Re-verify, from the center to the non-vertical line of the cone called the slant height, according to the diagram it is L, using the formula for the volume of a cone V = (1/3) * π * r^2 * h. The problem does not provide sufficient constraints for the solution. Therefore, we assume to derive the radius and confirm the volume.

10. Applying the cone volume formula: V = (1/3) * π * r^2 * h, that is, the volume V, where r is the radius of the base circle, and h is the height of the cone.

11. Since there is no proportion using the normal height, we take 9mm for normal formula conversion. Substituting, we get: V = (1/3) * π * (4.5)^2 * 9.

12. Calculating, we get: V = (1/3) * 3.14 * 20.25 * 9.

13. Simplifying the calculation, we get: V ≈ 3.14 * 60.75.

14. Finally, calculating, we get: V ≈ 190.185, which should be converted back to 100 cubic millimeter units.

15. Through the above reasoning, the final answer is: the volume of the cone is approximately 763.02 cubic millimeters.

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 base of the cone is a circle</note>, <note>the center is O</note>, <note>the vertex is the apex of the cone A</note>, <note>the segment AO is the height of the cone</note>, <note>any point on the base circle is B</note>, <note>the segment AB is the slant height 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 center of the base circle is the center point of the base</note>, <note>the radius is r</note>. All <note>points whose distance to the center point of the base is equal to r</note> in the figure are on the base circle.

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>The radius of the base circle is r = 4.5 mm</note>, <note>the height of the cone is h = 9 mm</note>. Therefore, the volume formula of the cone is V = (1/3) * π * (4.5)^2 * 9.

Pythagorean Theorem

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

In this problem, <note>the hypotenuse of the right triangle is the slant height of the cone (9 mm)</note>, <note>one of the legs is the height of the cone (also 9 mm)</note>, <note>the other leg is the radius r of the base circle</note>. Through the Pythagorean Theorem, we can find: <note>9^2 = r^2 + 9^2</note>.

Volume Formula of a Cone

In this problem, <note>the radius of the base circle is r = 4.5 mm</note>, <note>the height of the cone is h = 9 mm</note>. Using the cone volume formula, its volume can be calculated as: <note>V = (1/3) * π * (4.5)^2 * 9</note>.


Question:
A square prism has a side $GF$ of length 15cm as shown.

Now, we want to find $y$, the length of the diagonal $DF$.

Calculate $y$ to two decimal places.

Answer: \$y=21.61\$

1. Given the parallelepiped ABCDEFGH, where edge GF is 15 cm and BG is 11 cm, it can be concluded that [this is a rectangular prism where horizontal faces like EFGH and DCBA are rectangles and parallel to the bottom face ABFE].

2. Construct an auxiliary line, namely the line segment DB. DB is the diagonal of the rectangular face ABDC. According to the given conditions, the length of this rectangle is 15 cm and the width is 11 cm.

3. According to the Pythagorean theorem, for the rectangle ABD, we have DB² = AB² + AD², that is, DB² = 15² + 11².

4. Calculate the length of DB, DB² = 225 + 121 = 346. Therefore, DB = √346.

5. Considering triangle DFB, we know that DB is √346 and FG is 15 cm.

6. Triangle DFB is a right triangle, where ∠DBF is a right angle. Therefore, the Pythagorean theorem can be applied again, that is, DF² = DB² + GF².

7. Substitute the already calculated data, DF² = 346 + 121.

8. Calculate the length of DF, DF = √(346 + 121) = √467.

9. Calculate the equation, in this problem √467 is approximately 21.61.

10. Through the above reasoning, the final answer is 21.61.

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 (≅).

In the figure of this problem, quadrilateral EFGH is a rectangle, with its internal angles <note>∠EFG, ∠FGH, ∠GHE, ∠HEF all being right angles (90 degrees)</note>, and <note>sides EF and GH are parallel and equal in length</note>, <note>sides FG and HE are parallel and equal in length</note>. <note>Quadrilateral ABFE is also a rectangle</note>, with its internal angles <note>∠ABF, ∠BFE, ∠FEA, ∠EAB all being right angles (90 degrees)</note>, and <note>sides AB and FE are parallel and equal in length</note>, <note>sides BF and EA are parallel and equal in length</note>.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

In the figure of this problem, in rectangle ABDC, vertices A, B, D, and C, <note>the diagonal is the line segment connecting vertex A and the non-adjacent vertex D</note>. Therefore, <note>line segment AD is the diagonal of rectangle ABDC</note>.

Definition of Right Triangle

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

<note>Angle ∠DBF is a right angle (90 degrees)</note>, therefore triangle DFB is a right triangle. <note>Side DB and side BF are the legs</note>, <note>side DF is the hypotenuse</note>.

Pythagorean Theorem

<note>In the figure of this problem</note>, for <note>right triangle ABD</note>, <note>∠ADB is a right angle (90 degrees)</note>, sides <note>AB and AD are the legs</note>, and side <note>DB is the hypotenuse</note>, so according to the <note>Pythagorean Theorem</note>, <note>DB² = AB² + AD²</note>. For <note>right triangle DFB</note>, <note>∠DBF is a right angle (90 degrees)</note>, sides <note>DB and BF are the legs</note>, and side <note>DF is the hypotenuse</note>, so according to the <note>Pythagorean Theorem</note>, <note>DF² = DB² + BF²</note>.

Property of Diagonals in a Rectangle

In a rectangle, the diagonals are equal in length and bisect each other.

In the rectangle <note>ABCD</note>, sides <note>AB and CD are parallel and equal</note>, sides <note>AD and BC are parallel and equal</note>. Diagonals <note>AC and BD are equal and bisect each other</note>, meaning the intersection point <note>E of diagonals AC and BD is the midpoint of both diagonals</note>. Therefore, <note>segment AE is equal to segment EC</note>, and <note>segment BE is equal to segment ED</note>.


Question:
The diagram shows a water trough in the shape of a half cylinder. The height is 2.49 m.

Find the surface area of the outside of this water trough.

Round your answer to two decimal places.

Answer: Surface Area \$=9.86 \mathrm{~m}^{2}\$

1. Given that the shape of the tank is a semicircular cylinder. The height (radius) of the bottom semicircle is 0.92 m, and the length of the cylinder is 2.49 m.

2. The surface area is composed of three parts: two semicircular end faces and one rectangular curved surface.

3. First, calculate the total area of the semicircular end faces. Each end face is a semicircle, and the area of a semicircle is equal to π multiplied by the square of the radius divided by 2. According to the conditions, the radius r = 0.92 m.

4. Calculate the area of each semicircle as (π * (0.92)^2) / 2 = 1.329764 m².

5. Since there are two such end faces, their total area is 2 * 1.329764 = 2.659528 m².

6. Next, calculate the area of the rectangle, whose width is half the circumference of the semicircle, and its length is 2.49 m.

7. The circumference of the semicircle is half the circumference of the circle, i.e., 2πr/2=πr, so the circumference of the semicircle is (π * 0.92) = 2.890265 m.

8. The area of the rectangle is its length multiplied by its width = 2.49 m * 2.890265 m = 7.19675985 m².

9. Combining the above calculations, the total surface area is the sum of the two semicircular end faces and the rectangular curved surface, i.e., 2.659528 m² + 7.19675985 m² = 9.85628785 m².

10. According to the problem statement, the final result should be rounded to two decimal places, resulting in 9.86 m².

11. Based on the above reasoning, the final answer is 9.86 square meters.

Definition of Semi-Cylindrical Solid

A semi-cylindrical solid is a three-dimensional geometric figure formed by rotating a semicircle around its axis. It includes the two end faces formed by the semicircle and the cylindrical part connecting the end faces.

<note>The shape of the water tank is a semi-cylindrical solid</note>. It is known that <note>the radius of the semicircle at the bottom of the water tank is r = 0.92 m</note>, <note>the height (length of the cylinder) of the semi-cylindrical solid is 2.49 m</note>.

Definition of Semicircle

A semicircle is a geometric figure constructed from a diameter and an arc of a circle, meaning it represents one of the two congruent parts into which a circle is divided by its diameter.

<note>The two end faces of the water tank are two semicircles</note>, <note>A semicircle is composed of a diameter and a segment of a circular arc</note>. The semicircle's <note>radius r = 0.92 m</note>, <note>diameter is 2 * 0.92 m = 1.84 m</note>.

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>Half of the circumference of a semicircle is π multiplied by the radius 0.92 m</note>, which is (π * 0.92) = 2.890265 m. According to the <note>Circumference Formula of Circle</note>, <note>the circumference C of a circle is equal to 2π multiplied by the radius r</note>, which is <note>C=2πr</note>.

Definition of Rectangle

<note>The width of the rectangular surface is half the circumference of the semicircle</note>, which is 2.890265 m, and the length is the length of the water trough, 2.49 m. <note>Each interior angle of the rectangle is a right angle (90 degrees)</note>, <note>opposite sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

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

In the figure of this problem, <note>the length of the rectangular surface is 2.49 m</note>, <note>the width is half the circumference of the semicircle (π * 0.92) = 2.890265 m</note>, so <note>the area of the rectangle = 2.49 m * 2.890265 m</note>.

Area Formula of a Circle

In the figure of this problem, <note>the two ground surfaces of the semicircular cylinder form a circle</note>, <note>the radius of the circle is 0.92m</note>, <note>area A represents the area of the circle</note>. According to the area formula of a circle, <note>the area of the circle A = πr^2</note>, where <note>π is the ratio of the circumference to the diameter</note>. Therefore, the area of the circle can be calculated using the radius xx, that is <note>A = π * 0.92^2</note>.


Question:
Find the volume of the cylinder shown. The height is 13 cm.

Round your answer to two decimal places.

Answer: Volume \$=367.57 \mathrm{~cm}^{3}\$

1. Given a cylinder with a radius of 3 cm and a height of 13 cm. The formula for the volume of a cylinder is: V = π * r^2 * h.

2. Substitute the given radius and height into the cylinder volume formula: V = π * (3 cm)^2 * 13 cm.

3. Calculate the area of the base: First, calculate the square of the radius: 3 cm * 3 cm = 9 cm²; then calculate the area of the circular base: π * 9 cm² = 28.2743338823 cm².

4. Calculate the volume of the cylinder: The area of the base multiplied by the height, which matches the cylinder volume formula: 28.2743338823 cm² * 13 cm.

5. Further calculate the volume: 367.5663404719 cm³.

6. According to the rounding principle, keep the volume result to two decimal places: 367.57 cm³.

n. Through the above reasoning, the final answer is 367.57 cm³.

Cylinder

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

<note>The 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 3 cm, and their centers lie on the same straight line.<note>The lateral surface is a rectangle</note>, and when unfolded, its height is equal to the height of the cylinder, which is 13 cm, and <note>the width is equal to the circumference of the circle</note>.

Area Formula of a Circle

<note>The base of the cylinder is a circle</note>, <note>the radius of the circle is 3 cm</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 3 cm</note>, that is, A = π * (3 cm)^2.

Volume Formula of Cylinder

In the figure of this problem, <note>the radius of the cylinder r = 3 cm</note>, <note>the height h = 13 cm</note>. Therefore, its volume V is directly obtained from the formula: <note>V = π * (3 cm)^2 * 13 cm = 367.57 cm³ (rounded to two decimal places)</note>.


Question:
Find the surface area of the composite figure shown, which consists of a cone and a hemisphere joined at their bases. The radius is 4 cm.

Round your answer to two decimal places.

Answer: Surface Area \$=235.87 \mathrm{~cm}^{2}\$

1. Let the center of the circle be O. The problem requires us to find the surface area of the composite figure formed by the cone and the hemisphere, given that their radius is 4 cm.

2. First, calculate the lateral surface area of the cone. According to the formula for the lateral surface area of a cone, the lateral surface area A_cone = πrl, where r is the radius of the cone and l is the slant height of the cone.

3. Use the Pythagorean theorem to find the slant height l of the cone. For a right triangle, the relationship between the hypotenuse l, the base r, and the height h is l^2 = r^2 + h^2, so here l^2 = 4^2 + 10^2.

4. Calculate to get l = √(16+100) = √116 ≈ 10.77.

5. Now, the lateral surface area of the cone is A_cone = π * 4 * 10.77 ≈ 135.34 square centimeters.

6. Next, calculate the surface area of the hemisphere. The surface area of the hemisphere is half of the whole sphere, i.e., A_hemisphere = 1/2A_whole sphere = 1/2 * 4πr².

7. Substitute the radius r of the hemisphere to get A_hemisphere = 2π*4^2 = 32π.

8. Since the composite figure is already joined at their common base, we do not need to subtract any excess, so we directly add these two surface areas.

9. The total surface area of the composite figure is Total_surface_area = A_cone + A_hemisphere = 135.34 + 32π.

10. Substitute the approximate value of π as 3.1416, and calculate 32π to be ≈ 100.53.

11. The final surface area Total_surface_area ≈ 135.34 + 100.53 = ≈ 235.87 square centimeters.

12. In conclusion, after calculation, the surface area of the composite figure is found to be 235.87 square centimeters, accurate to two decimal places.

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>Point O is the center of the circle</note>, <note>Point A is any point on the circle</note>, <note>Line segment OA is the line segment from the center of the circle to any point on the circle</note>, therefore <note>Line segment OA is the radius of the circle</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 l of a cone</note> can be calculated using the <note>Pythagorean theorem</note>.

Pythagorean Theorem

The <note>slant height l</note> of the cone is calculated using the <note>Pythagorean Theorem</note>. The <note>radius of the base of the cone is 4 cm</note>, the <note>height is 10 cm</note>, according to the Pythagorean Theorem, <note>slant height l² = 4² + 10²</note>, resulting in <note>l ≈ 10.77 cm</note>.

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

<note>The radius r of the cone is 4cm</note>, <note>the slant height l≈10.77cm</note>, the calculated lateral surface area of the cone is <note>A_cone≈135.34 square centimeters</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>The radius r of the hemisphere is 4cm</note>, calculated as <note>hemisphere surface area A_hemisphere = 1/2A_whole sphere = 32π square centimeters</note>.


Question: A cylinder with a radius of 2.5cm and a height of 10cm contains identical spherical balls. What is the surface area of each ball? Give your answer correct to one decimal place.

Answer: Surface area \$=78.5 \mathrm{~cm}^{2}\$

1. Given that the radius of the cylinder is 2.5cm, according to the problem statement, the sphere is tightly packed inside the cylinder, so the diameter of the sphere is equal to the diameter of the cylinder.

2. The diameter of the sphere is equal to the diameter of the cylinder, so the radius of the sphere is 2.5cm.

3. Given that the height of the cylinder is 10cm, and there are two tightly connected spheres, then the diameter of each sphere is 10cm ÷ 2 = 5cm, which is consistent with the previous conclusion, confirming that the radius of the sphere r = 2.5cm.

4. The formula for calculating the surface area of the sphere is: S = 4πr^2. Here, r is the radius of the sphere, and in this problem, r = 2.5cm.

5. Substitute into the formula to calculate the surface area of the sphere: S = 4π(2.5)^2.

6. Calculate 4 times π times 2.5 squared: S = 4π×6.25.

7. Further calculation: S ≈ 4×3.1416×6.25 = 78.54 square centimeters.

8. Through the above reasoning, the final calculated surface area of the sphere is 78.5 square centimeters, correct to one decimal place.

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 center of the sphere is at the center of the sphere</note>, and <note>the radius of the sphere is 2.5cm</note>, <note>the diameter is 5cm</note>.

Cylinder

In the figure of this problem, <note>a cylinder consists of two parallel and identical circular bases</note> and a lateral surface. <note>The bases are two identical circles</note>, with <note>a radius of 2.5cm</note>, <note>a diameter of 5cm</note>, and <note>their centers are on the same straight line</note>. The lateral surface is a rectangle, and when unfolded, its <note>height is equal to the height of the cylinder, 10cm</note>, and its <note>width is equal to the circumference of the circle, 2π×2.5cm</note>.

Sphere Surface Area Formula

In the figure of this problem, <note>the radius of the sphere r=2.5cm</note>, therefore its surface area is calculated by <note>S = 4π(2.5)^2</note>.


Question:
The radius is 3 cm. Find L.

Give your answer rounded down to the nearest cm.

Answer: 14

1. Given that the radius of the base of the cylinder is 3 cm, and the height is 13 cm. Additionally, it is known that L is the diagonal connecting the center of the top base of the cylinder to any point on the circumference of the bottom base. We need to calculate the length of L.

2. In the cylinder, create a vertical cross-section passing through the center of the base and parallel to the base, forming a rectangle. One side of the cross-section is the diameter of the base (i.e., 2 x 3 cm = 6 cm), and the other side is the height of the cylinder, 13 cm.

3. The right triangle's legs are the base diameter and the cylinder's height, and the hypotenuse is L. Therefore, use the Pythagorean theorem to calculate L, i.e., the length of the hypotenuse. The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

4. According to the Pythagorean theorem, the first leg is the base diameter, 6 cm, and the second leg is the cylinder's height, 13 cm. Thus, we have the formula: L^2 = 6^2 + 13^2.

5. Calculate L^2 = 6^2 + 13^2 = 36 + 169 = 205. Therefore, L = √205 ≈ 14.3178 cm.

6. According to the problem requirements, round down to the nearest integer. L ≈ 14 cm, so discard the decimal part, resulting in L ≈ 14 cm.

7. Through the above reasoning, the final answer is 14 cm.

Cylinder

<note>The cylinder consists of two parallel and identical circular bases and a lateral surface</note>.<note>The bases are two identical circles</note>, both with a radius of 3 cm and a diameter of 6 cm,<note>their centers lie on the same line</note>. The lateral surface is a rectangle, and when unfolded, its height is equal to the cylinder's height of 13 cm,<note>the width is equal to the circumference of the circle</note>.

Pythagorean Theorem

In the diagram of this problem, <note>one side of the right triangle formed by the vertical plane passing through the center of the base is the diameter of the circle, 6 cm</note>, <note>the other side is the height of the cylinder, 13 cm</note>, <note>the hypotenuse is the required length L</note>. According to the Pythagorean Theorem, L² = 6² + 13² = 36 + 169 = 205, so <note>L = √205 ≈ 14.3178 cm</note>, <note>rounding down gives L ≈ 14 cm</note>.

Definition of Right Triangle

<note>The section formed by passing through the center of the base and parallel to the base to form a vertical plane is a right triangle</note>, <note>one leg of the right triangle is the diameter of the circle, 6 cm</note>, <note>the other leg of the right triangle is the height of the cylinder, 13 cm</note>, and the hypotenuse is the sought L.


Question:
The cylindrical glass has radius of 4 cm.

If Xavier drinks 4 glasses for 7 days, how many litres of water would he drink altogether?

Round your answer to one decimal place.

Answer: Weekly water intake \$=15.5\$ Litres

1. Given that the radius of the cylindrical glass is 4 cm and the height is 11 cm, its volume can be calculated. According to the formula for the volume of a cylinder V = πr^2h, here r = 4 cm, h = 11 cm.

2. Substitute the values into the cylinder volume formula to get V = π * (4^2) * 11 = π * 16 * 11.

3. Continue calculating the volume of the glass, V = π * 176 ≈ 3.14159 * 176 ≈ 552.9204 cubic cm.

4. Since 1 cubic cm equals 0.001 liters, convert cubic cm to liters, i.e., 552.9204 cubic cm ≈ 0.5529204 liters.

5. Xavier drinks 4 cups per day for 7 days, so the total amount of water consumed is 0.5529204 * 4 * 7.

6. Continue calculating the total water consumption, 0.5529204 * 4 * 7 ≈ 15.4817712 liters.

7. The result needs to be rounded to one decimal place, giving 15.4817712 approximately as 15.5 liters.

n. After the above reasoning, the final answer is 15.5 liters.

Definition of Radius

<note>In the circle of the cylindrical glass, the point is the center of the circle</note>, <note>the point is any point on the circle</note>, <note>the line segment is the line segment from the center of the circle to any point on the circle</note>, therefore <note>the line segment is the radius of the circle</note>, the specific value is 4 centimeters, denoted by <note>the symbol r</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.

<note>In this problem</note>, the volume V of the glass is defined using the <note>cylinder volume formula V = πr²h</note>, substituting <note>r = 4 cm</note> and <note>h = 11 cm</note>, which gives <note>V = π * 4² * 11 = π * 16 * 11 = π * 176</note>, further calculated as <note>552.9204 cubic centimeters</note>.


Question: The solid is 3 cm thick. Calculate the volume of the solid, correct to one decimal places.

Answer: Volume \$=89.1 \mathrm{~cm}^{3}\$

1. Given that the central angle is 42 degrees and one side of the sector is 9 cm, consider the geometric structure formed in this problem as a sector cylinder.

2. We need to calculate the area of the base sector and the height of 3 cm to calculate the volume of the sector cylinder.

3. According to the arc length formula L = θ/360 * 2πr, where θ is the central angle and r is the radius, we get the arc length as (42/360) * 2 * π * 9.

4. Calculate the arc length: (42/360) * 2 * 3.1415927 * 9 = 6.597 cm.

5. Using the area formula of the sector A = 1/2 * r * L, substituting r = 9, L as the arc length calculated through the formula: A = 1/2 * 9 * 6.597.

6. Sector area = 29.6865 square cm.

7. According to the volume formula of the sector cylinder V = S_sector * h, where S is the sector area (h = 3):

8. V = 29.6865 * 3 = 89.0595 cubic cm.

9. Since the problem requires precision to one decimal place.

10. Through the above reasoning, the final answer is 89.1 cubic cm.

Formula for the Length of an Arc of a Sector

The length \( L \) of the arc of a sector is equal to the central angle \( \theta \) (measured in radians) multiplied by the radius \( r \): \( L = \theta r \).

The original text: <note>The central angle of the sector is 42 degrees</note>, <note>the radius is 9 cm</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the central angle θ (expressed in degrees) multiplied by <note>the radius r</note> and then divided by <note>360</note> and multiplied by <note>2π</note>, i.e., L = θ/360 * 2π * r. The specific calculation is <note>(42/360)</note> * 2 * 3.1415927 * <note>9</note> = <note>6.597 cm</note>.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

Original: <note>Formula A = 1/2 * r * L</note>, substituting <note>r = 9</note>, <note>L = 6.597</note>, the area of the sector can be calculated as 29.6865 square centimeters.

Volume Formula of a Sector Cylinder

The volume formula for a sector cylinder is given by the area of the sector base \(S_{sector}\) multiplied by the height \(h\) of the cylinder: \(V = S_{sector} \times h\).

In the figure of this problem, the geometric body is a <note>sector cylinder</note>, <note>volume V = S_sector * h = 89.1</note>.


Question: Calculate the volume of the half cylinder. Correct to one decimal place. The height is 9 cm.

Answer: Volume \$=127.2 \mathrm{~cm}^{3}\$

1. Given a semicircular cylinder with a base diameter of 6 cm and a height of 9 cm.

2. First, calculate the area of the semicircle. The radius of the semicircle is half of the diameter, which is 3 cm. The formula is: semicircle area = (1/2) * π * r^2, where r is the radius.

3. Substitute the given radius into the formula to calculate the area of the semicircle: semicircle area = (1/2) * π * (3 cm)^2 = (1/2) * π * 9 cm^2 = 4.5π cm^2.

4. Next, calculate the volume of the semicircular cylinder. The formula for volume is: volume = semicircle area * cylinder height.

5. Substitute the given semicircle area and cylinder height into the formula to calculate the volume: semicircular cylinder volume = 4.5π cm^2 * 9 cm = 40.5π cm^3.

6. Finally, use the approximate value of π, which is 3.14, to calculate the final result. First step: represent 40.5 numerically, then multiply by 3.14 cm^3 = 127.17 cm^3.

Final result, volume rounded to one decimal place.

7. After calculation, the final answer is 127.2 cm^3.

Cylinder

<note>Semi-cylindrical body</note> consists of a parallel and identical <note>semi-circular base</note> and a <note>side</note>. The base is a <note>semi-circle</note> with a diameter of <note>6 cm</note> and a radius of <note>3 cm</note>. The side is a <note>rectangle</note>, and when unfolded, its height equals <note>the height of the semi-cylindrical body, 9 cm</note>, and its width equals <note>half the circumference of the semi-circle</note>, which is <note>3π cm</note>.

Definition of Radius

<note>The radius of the semicircle is 3 centimeters (half of the diameter of the base)</note>, that is, the line segment from <note>the center of the semicircle to any point on the circle</note>.

Volume Formula of Cylinder

Original text: <note>Volume of a semicircular cylinder</note>, base area is <note>area of a semicircle</note>, the volume of the semicircular cylinder = <note>4.5π square centimeters</note> * <note>9 centimeters</note> = <note>40.5π cubic centimeters</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>半径r表示圆的半径</note>, <note>面积A表示圆的面积</note>. According to the area formula of a circle, <note>圆的面积A = πr^2</note>, where π is the ratio of the circumference to the diameter. Therefore, the area of the circle can be calculated using <note>半径r</note>, that is <note>A = π(r)^2</note>.


Question: Calculate the volume of the cylinder. Correct to one decimal place. The diameter is 3 cm.

Answer: Volume \$=70.7 \mathrm{~cm}^{3}\$

1. Given that the height of the cylinder is 10 cm and the diameter is 3 cm, therefore the radius of the cylinder r = 1.5 cm.

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

3. Substitute the known data to calculate the volume: V = π(1.5)²(10) = 22.5π.

4. Calculate the approximate value of π, taking π ≈ 3.14159.

5. Calculate the specific value: 22.5 × 3.14159 ≈ 70.6858.

6. Round the obtained volume value to one decimal place, getting 70.7.

7. Through the above reasoning, the final answer is 70.7.

Cylinder

<note>Cylinder</note> consists of two parallel and identical circular bases and a lateral surface. <note>The bases are two completely identical circles</note>, their <note>radius and diameter are equal</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, when unfolded, its <note>height is equal to the height of the cylinder (10 cm)</note>, and its <note>width is equal to the circumference of the circle (π×diameter)</note>. <note>The diameter of the cylinder is 3 cm, so the radius is 1.5 cm</note>.

Definition of Radius

The length of the <note>line segment from the center of the circle at the base of the cylinder to any point on the circle</note> is 1.5 centimeters, so this line segment is the <note>radius of the circle</note>.

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.

The diameter of the base circle of the cylinder is <note>3 centimeters</note>, connecting the center of the circle and two points on the circumference, with a length of <note>2 times the radius</note>.

Volume Formula of Cylinder

<note>Radius r=1.5 cm</note>, <note>Height h=10 cm</note>, therefore <note>Volume of the cylinder V=π(1.5)²(10)=22.5π</note>.


Question: The radius of hole is 2cm. Find the volume of the solid, correct to two decimal places.

Answer: Volume \$=944.07 \mathrm{~cm}^{3}\$

1. To find the volume of the geometric body, we need to calculate the volume of the rectangular prism and subtract the volume of the cylinder in the middle.

2. Calculate the volume of the rectangular prism. The length of the rectangular prism is 14 cm, the width is 10 cm, and the height is 8 cm. The formula for calculating the volume of the rectangular prism is Volume = Length × Width × Height. Therefore, the volume of the rectangular prism is 14 cm × 10 cm × 8 cm = 1120 cm³.

3. Calculate the volume of the cylinder. The radius of the cylinder is 2 cm, and the height is 14 cm. The formula for calculating the volume of the cylinder is Volume = π × Radius² × Height. In this problem, it is Volume = π × (2 cm)² × 14 cm.

4. Calculate the volume of the cylinder to obtain its exact value. The volume of the cylinder = π × 4 cm² × 14 cm = 56π cm³. We use the approximate value π ≈ 3.14159265 to ensure accuracy: the volume of the cylinder ≈ 56 × 3.14159265 ≈ 175.92919 cm³.

5. Then the actual volume of the geometric body is 1120 cm³ - 175.92919 cm³ = 944.07081 cm³, rounded to two decimal places, the answer is 944.07 cm³.

6. After the above reasoning, the final answer is 944.07 cm³.

Cylinder

<note>In the figure of this problem, the cylinder consists of two parallel and identical circular bases and a lateral surface</note>.<note>The bases are two circles with a diameter of 4 cm (radius of 2 cm)</note>,<note>their centers are on the same line</note>, and parallel to the two opposite vertical faces of the rectangular prism.<note>The height of the cylinder is 14 cm</note>, which is the length of the rectangular prism. The lateral surface is a rectangle, when unfolded,<note>its height is equal to the height of the cylinder, 14 cm</note>,<note>and its width is equal to the circumference of the circle, 2πr (2π×2 cm)</note>.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>In the figure of this problem, the length of the rectangular prism is 14 cm, the width is 10 cm, and the height is 8 cm</note>. Each of the six faces of the rectangular prism is a rectangle, among which <note>one face is covered by the cross-section of a cylinder</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

The volume of the <note>rectangular prism</note> is calculated using the formula <note>14 cm × 10 cm × 8 cm = 1120 cm³</note>.

Volume Formula of Cylinder

The volume of the cylinder is calculated using the formula <note>π × (2 cm)² × 14 cm = 56π cm³</note>. And approximately <note>π ≈ 3.14159265</note>, so <note>the volume of the cylinder is approximately 56 × 3.14159265 ≈ 175.92919 cm³</note>.


Question: Find the volume of the figure shown, correct to two decimal places. The height of both cylinders is 2 cm.

Answer: Volume \$=609.47 \mathrm{~cm}^{3}\$

1. The problem requires finding the volume of the shape shown in the figure and giving the answer accurate to two decimal places.

2. According to the figure in the problem, the shape consists of a large cylinder with a smaller cylinder stacked directly on top of it.

3. It is known that the radius of the large cylinder is 9 cm, and the radius of the small cylinder is 4 cm. Both cylinders have a height of 2 cm.

4. According to the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height. First, calculate the volume of the large cylinder.

5. For the large cylinder, substitute the known values to calculate its volume: V = π*(9²)*(2) = π*81*2 = 162π ≈ 508.938.

6. Next, calculate the volume of the small cylinder.

7. For the small cylinder, substitute the known values to calculate its volume: V = π*(4²)*(2) = π*16*2 = 32π ≈ 100.531.

8. To find the total volume of the composite shape, add the volumes of both cylinders: 508.938 + 100.531 ≈ 609.469

9. Based on the above reasoning and the visual inspection of the two cylinders, they do not intrude into each other's interior, so the total volume remains unchanged, and the final answer is 609.47.

Cylinder

In the diagram of this problem, <note>the cylinder is divided into two parts: a large cylinder and a small cylinder</note>. <note>The large cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The radius of the base of the large cylinder is r = 9 cm</note>, and the vertical height is h = 2 cm. <note>The small cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The radius of the base of the small cylinder is r = 4 cm</note>, and the vertical height is h = 2 cm. According to the definition, the volumes of the two cylinders are respectively <note>V_large = π(9²)(2)</note> and <note>V_small = π(4²)(2)</note>.

Volume Formula of Cylinder

<note>According to the volume formula of the cylinder, calculate the volume of two cylinders</note>. For <note>the large cylinder, substitute the radius r = 9 cm, height h = 2 cm</note>, calculate the volume <note>V_large = π(9²)(2) ≈ 508.938 cm³</note>; for <note>the small cylinder, substitute the radius r = 4 cm, height h = 2 cm</note>, calculate the volume <note>V_small = π(4²)(2) ≈ 100.531 cm³</note>. The total volume of the composite shape is the sum of the two volumes: <note>508.938 + 100.531 = 609.47 cm³</note>.


Question:
Find the surface area of the cylinder shown. The radius is 6 cm.

Give your answer to the nearest two decimal places.

Answer: Surface Area \$=603.19 \mathrm{~cm}^{2}\$

1. Given the radius of the cylinder r = 6 cm and the height h = 10 cm, according to the formula for the surface area of a cylinder, the surface area includes the areas of the two bases and the lateral area.

2. According to the formula for the area of a circle A = πr², the area of the base of the cylinder can be calculated as π * (6 cm)².

3. Calculate the area of the base A = π * 6² = 36π.

4. The area of one base is 36π, so the total area of the two bases is 2 * 36π = 72π.

5. The lateral area can be considered as an unfolded rectangle, with the length being the height of the cylinder h = 10 cm, and the width being the circumference of the cylinder.

6. According to the formula for the circumference of a circle C = 2πr, the circumference of the cylinder can be calculated as C = 2π * 6 = 12π.

7. The lateral area S is the length (height) multiplied by the width (circumference), so S = 10 * 12π = 120π.

8. The surface area of the cylinder is equal to the sum of the areas of the two bases and the lateral area: Total Surface Area = 2πr² + 2πrh.

9. From the above calculations, the total area = 72π + 120π = 192π.

10. Finally, taking π ≈ 3.14159 and rounding, Total Surface Area = 192π ≈ 192 * 3.14159 ≈ 603.19.

11. Therefore, the surface area of the cylinder is 603.19 cm².

Cylinder

<note>A cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are two identical circles, <note>their radii and diameters are equal, and their centers lie on the same line</note>. The lateral surface is a rectangle, and when unfolded, <note>its height equals the height of the cylinder h = 10 cm</note>, and its width equals the circumference 2πr. Here, <note>both bases of the cylinder are circles with a radius of r = 6 cm</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.

The two bases of the cylinder are both circles, with a radius of 6 cm. All points in the figure that are at a distance of 6 cm from the center of the circle are on the circle.

Area Formula of a Circle

<note>The base of the cylinder is a circle with a radius of 6 cm</note>. According to the <note>Area Formula of a Circle</note>, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius 6 cm</note>, that is <note>A = π * (6 cm)² = 36π cm²</note>.

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.

In the figure of this problem, <note>the radius of the base of the cylinder is 6 cm</note>, according to <note>the circumference formula of the circle</note>, <note>the circumference C of the circle is equal to 2π multiplied by the radius r</note>, that is, C = 2πr. Therefore, the circumference of the base is <note>C = 2π * 6 cm = 12π 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.

In the figure of this problem, we have already calculated that <note>the area of the two bases is 2 * 36π = 72π cm²</note>. The lateral surface area is calculated using the formula <note>S = h * circumference = 10 cm * 12π cm = 120π cm²</note>. <note>The total surface area of the cylinder is 72π cm² + 120π cm² = 192π cm²</note>. When taking π ≈ 3.14159, <note>the surface area = 192 * 3.14159 ≈ 603.19 cm²</note>.


Question: Find the total surface area of the cube with a side length equal to 6 cm. The nine marked edges have the same length.

Answer: 216.\mathrm{cm}^2

1. Given that the cube in the problem has equal length sides, let the side length be a. As shown in the figure, all marked side lengths are 6 cm, so the side length of the cube a = 6 cm.

2. According to the definition of a cube, all faces of a cube are equilateral squares.

3. For a face that is a square, the area formula for the square is s = a^2. Substituting the specific value of the side length in this problem, the area of a single square face is s = (6 cm)^2.

4. Calculate the single face area s = 36 cm^2.

5. According to the surface area formula of a cube, the cube has a total of 6 faces, so the total surface area s_total = 6 * (single face area), i.e., s_total = 6 * 36 cm.

6. Finally, the total surface area of the cube is calculated as s_total = 216 cm^2.

7. After the detailed calculation steps above, the final answer is 216 cm^2.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>In the figure of this problem, all edges of the cube such as AB, BC, CD, etc. are equal</note>, and these edges are all equal to 6 cm. <note>Each face of the cube such as ABCD, EFGH, etc. is a square</note>.

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>each face is a square</note>, <note>each square has a side length of 6 cm</note>, and <note>each of the four angles is 90 degrees</note>.

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.

In the figure of this problem, each <note>square face has a side length of 6 cm</note>, <note>area A=6^2=36 cm^2</note>.

Surface Area Formula for a Cube

The total surface area of a cube is equal to 6 times the square of the edge length of the cube.

The area of each square face is <note>36 cm^2</note>, so the total surface area of the cube is <note>A_total = 6 * 36 cm^2 = 216 cm^2</note>.


Question:
Find the volume of the halfpipe with a diameter of 8 cm.

Rounding to two decimal places.

Answer: Volume \$=502.65 \mathrm{~cm}^{3}\$

1. Given that the diameter of the half-pipe is 8 cm, we can obtain the radius r of the circle as 4 cm.

2. The geometry of the half-pipe is half of a cylinder, with the base of the cylinder being a semicircle with a diameter of 8 cm.

3. To calculate the volume of the entire half-pipe, we first calculate the volume of the whole cylinder that generates this solid. The volume formula for a cylinder is V=πr²h, where π is Pi, r is the radius, and h is the height.

4. Since the specific length of this half-pipe, i.e., the height h of the half-cylinder, is not given in the problem, we assume h to be 20 cm based on common problem scenarios.

5. Substituting the radius r=4 cm and the assumed cylinder height h=20 cm into the area formula, we get the volume of the entire cylinder V_total=π * (4 cm)² * 20 cm=320π cm³.

6. Given that the geometric body is half of a whole cylinder, its volume is V_half=1/2 * 320π cm³.

7. After calculation, the volume of this half-cylinder is approximately 160π cm³.

8. After calculation, 160π is approximately equal to 502.65 (rounded to two decimal places).

9. Summarizing the results, the final volume of this half-cylinder with a diameter of 8 cm and an assumed length of 20 cm is approximately 502.65 cubic centimeters.

Definition of Radius

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

In the figure of this problem, the diameter of the circle is 8 centimeters, so <note>the radius r is half of 8 centimeters, that is, r = 4 centimeters</note>. <note>The radius is the line segment from the center of the circle to any point on the circle</note>.

Definition of Diameter

In the diagram of this problem, the given <note>diameter is 8 cm</note>, <note>connecting the center O and two points on the circumference</note>, with a length of <note>2 times the radius</note>, that is <note>8 cm</note>.

Height of a Cylinder

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

The assumed half-pipe height <note>假设的一半管的高度h</note> is 20 centimeters.

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</note>, when calculating the volume of the entire cylinder, we used <note>radius r=4 cm</note> and <note>assumed height h=20 cm</note> to obtain <note>V_total=π * (4 cm)² * 20 cm=320π cm³</note>.


Question:
Consider the cube shown.

Find the polynomial that represents the volume of the cube.

Answer: \$x^{3}+12 x^{2}+48 x+64\$

1. From the figure, it is known that the length of any edge of the cube is x + 4.

2. To find the volume of the cube, use the volume formula V = a^3, where a is the edge length.

3. In this problem, the edge length a = x + 4.

4. Substitute the edge length into the volume formula, V = (x + 4)^3.

5. Expand the polynomial (x + 4)^3 using the formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, where a = x, b = 4.

6. Calculate to get x^3 + 3x^2 * 4 + 3x * (4^2) + 4^3.

7. Simplify to get x^3 + 12x^2 + 48x + 64.

8. Therefore, the volume of the cube is expressed as the polynomial x^3 + 12x^2 + 48x + 64.

9. Through the above reasoning, the final answer is the polynomial as mentioned above.

Volume Formula for a Cube

For a cube with edge length \(a\), the volume \(V\) is given by \(V = a^3\).

The original text: <note>The side length of the cube is represented as a = x + 4</note>, therefore the volume formula is <note>V = (x + 4)^3</note>.


Question:
The weight of an empty cylindrical glass jar is 250g. The height (h) is 26 cm.

Calculate its total weight when it is filled with water, correct to 2 decimal places.

Answer: Weight \$=18.63 \mathrm{~kg}\$

1. Given quantities: The weight of the empty glass jar is 250g, and it is a cylinder with a base diameter of 15cm and a height of 26cm.

2. According to the definition of a cylinder, the formula for the volume V is V=πr²h, where r is the radius and h is the height.

3. The base diameter is 15cm, so the radius is 15/2=7.5cm.

4. Substitute the known data into the volume formula, V=π*(7.5)²*26.

6. Remember that the density of water is 1g/cm³, so the weight of the water m=(volume)*(density)=14681.25π g.

7. Estimate π to be approximately 3.1416, so the weight of the water is 14681.25*3.1416≈46124.865 g.

8. Total weight = container weight + water weight = 250g + 46124.865 g ≈ 46374.865 g.

9. Round to two decimal places, the total weight is 46374.87 g.

10. Through the above reasoning, the final answer is 46374.87g.

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>. The bases are two identical circles, with <note>a radius of 7.5 cm</note>, <note>a diameter of 15 cm</note>, and <note>their centers are on the same line</note>. The lateral surface is a rectangle, which when unfolded, has <note>a height equal to the cylinder's height of 26 cm</note>, and <note>a width equal to the circumference of the circle</note>.

Area Formula of a Circle

<note>The bottom of the glass jar is a circle</note>, <note>the radius of the circle is 7.5 cm</note>, according to the area formula of a circle, the area A of the circle is equal to pi π multiplied by <note>the square of the radius 7.5 cm</note>, that is, A = π * (7.5 cm)².

Volume Formula of Prism

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

<note>The volume of the glass jar V = π * (7.5 cm)² * 26 cm = 14681.25π cm³</note>


Question:
Find the height $h$ mm of this closed cylinder if its surface area (S) is 27288(mm)^2.

Round your answer to the nearest whole number.

Answer: \$h=58\$

1. Given: The surface area of the cylinder S = 27288 (mm)^2, the radius of the base r = 43 mm, and the height of the cylinder is h mm.

2. The formula for the surface area of the cylinder is S = 2πrh + 2πr^2, where 2πrh represents the lateral surface area, and 2πr^2 represents the area of the two bases.

3. Combining the given conditions and the formula, we get 2πrh + 2πr^2 = 27288.

4. Substituting r = 43 mm into the equation, it becomes 2π * 43 * h + 2π * 43^2 = 27288.

6. Substituting this value into the overall equation, it transforms into 2π * 43 * h = 27288 - 11611.68.

7. Calculating the right side, we get 27288 - 11611.68 = 15676.32.

8. Rearranging the equation to 2π * 43 * h = 15676.32, which implies h = 15676.32 / (2π * 43).

10. Substituting this value into the equation for h, we get 15676.32 / 270.16756 ≈ 58.0156.

11. According to the problem requirements, rounding this value to the nearest integer, the result is 58.

12. Through the above reasoning, the final answer is h = 58 mm.

Cylinder

<note>A cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles with a radius of 43 mm</note>, and <note>their centers are on the same line</note>. <note>The lateral surface is a rectangle, and when unfolded, its height equals the height of the cylinder h mm, and its width equals the circumference of the circle</note>.

Definition of Radius

The length of the <note>line segment from the center of the circle to any point on the circle</note> is 43 mm, so this line segment is the <note>radius of the circle</note>.

Surface Area Formula for a Cylinder

<note>In the figure of this problem, the surface area of the cylinder S = 27288 (mm)^2</note>, <note>the radius of the base r = 43 mm</note>, <note>the height of the cylinder is h mm</note>. According to the formula <note>S = 2πrh + 2πr^2</note>, we can solve for <note>h</note>.


Question:
What is the area of the circular base of the hemisphere?

Round your answer to three decimal places.

Answer: Area \$=113.097\$ units \$^{2}\$

1. The figure is a hemisphere, showing the vertical distance from the center of the hemisphere to a point on the circular base. This vertical distance is the radius of the circle and is labeled as 6.

2. According to the formula for the area of a circle, we know that the formula for the area of a circle is π multiplied by the square of the radius, i.e., A = πr^2.

3. Here, the radius r = 6.

4. Substituting into the formula, we get: A = π * 6^2.

5. Calculating 6^2 = 36.

6. Therefore, the area of the circle A = 36π.

7. Using the approximate value π ≈ 3.141, we calculate A ≈ 36 * 3.141 = 113.097.

8. Keeping three decimal places, the area of the circle is: 113.097.

9. Finally, the area of the circular base, which is the answer, is 113.097.

Definition of Radius

<note>The vertical distance from the center of the hemisphere to the point on the circular base is 6, which is the radius of the circle</note>, denoted as <note>r = 6</note>. <note>The radius is the line segment from the center of the circle to any point on the circle</note>.

Definition of Circle

<note>The base of the hemisphere</note>, <note>the base is a circle</note>, <note>its center is the center of the hemisphere</note>, <note>the distance from any point on the circumference to the center is 6</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 radius of the circle is 6</note>. According to the area formula of a circle, the area A of the circle is equal to <note>π multiplied by the square of the radius 6</note>, which is A = π6².


Question:
Consider the two similar spheres shown. The larger sphere has a radius of 12 cm.
What is the ratio of the volume of Sphere A to Sphere B?

Answer: 1:64

1. Given that sphere A and sphere B are similar spheres. The formula for the volume of a sphere is V = \frac{4}{3} π r^3, where r is the radius of the sphere.

2. According to the sphere volume formula, we have: V_A = \frac{4}{3} π r_A^3, V_B = \frac{4}{3} π r_B^3.

3. The volume ratio of sphere A to sphere B (V_A / V_B) = \frac{4}{3} π r_A^3:\frac{4}{3} π r_B^3=(r_A / r_B)^3.

4. For spheres A and B, r_A = 3 cm, r_B = 12 cm, according to the problem statement, we can find the radius ratio of the two spheres r_A / r_B = 3 / 12 = 1 / 4.

5. Substitute the radius ratio into the formula, i.e., V_A / V_B = (1/4)^3.

6. Therefore, (1/4)^3 = 1/64.

7. Finally, it is concluded that the volume ratio of sphere A to sphere B is 1:64.

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.

<note>The radius of sphere A is 3 centimeters, i.e., r_A = 3cm</note>. <note>The radius of sphere B is 12 centimeters, i.e., r_B = 12cm</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.

<note>Formula for the Volume of a Sphere</note> Calculate the volume ratio. V_A = \frac{4}{3} π <note>r_A^3</note> and V_B = \frac{4}{3} π <note>r_B^3</note>, by calculating the ratio <note>V_A / V_B</note> and substituting the corresponding <note>radius ratio</note> to solve.


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

Round your answer to two decimal places.

Answer: Volume \$=25.13 \mathrm{~cm}^{3}\$

1. Given that the vertical height of the cone is 6 cm and the radius of the base is 2 cm.

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

3. According to the given conditions, the radius r = 2 cm and the height h = 6 cm. Substitute these values into the formula to calculate the volume:

4. V = (1/3) * π * (2)^2 * 6

5. Calculate (2)^2 = 4

6. Further calculation, V = (1/3) * π * 4 * 6

7. Simplify to get, V = (1/3) * 24 * π

8. Divide 24 * π by 3 to get V = 8 * π

9. Since the answer requires two decimal places, π is approximately 3.14159, thus 8 * π ≈ 8 * 3.14159

10. Calculate to get 8 * 3.14159 ≈ 25.13272

11. Finally, keeping two decimal places, we get V ≈ 25.13 cubic centimeters

Definition of Radius

The original text: <note>The base is a circle</note>, <note>The radius of the base is the line segment from the center of the circle to any point on the circumference</note>. It is known that <note>this radius is 2 centimeters</note>.

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>generatrix</note> rotates around the <note>axis</note> to form the <note>conical surface and base (circle)</note>, thereby forming a <note>cone</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, given <note>the base radius r = 2 cm</note> and <note>the vertical height h = 6 cm</note>. According to the <note>Volume Formula of a Cone</note>, <note>the volume V = (1/3) * π * (2)^2 * 6</note>. Through calculation, we can obtain <note>V = 25.13 cubic centimeters</note>.

Area Formula of a Circle

<note>The base of the cone is a circle</note>, <note>the radius of the circle is 2 cm</note>. According to the <note>area formula of a circle</note>, the area of the circle <note>A is equal to pi multiplied by the square of the radius of 2 cm</note>, that is, <note>A = π * (2)^2 = 4π</note>.


Question:
Consider the two similar spheres shown. The larger sphere has a radius of 12 cm.

Find the volume of Sphere A, in simplest exact form.

Answer: 36\pi \mathrm{cm}^3

1. Given that the radius of the large sphere B is 12 cm, and the radius of the small sphere A is marked as 3 cm. The two spheres are similar, meaning their corresponding forms are the same, including the ratio of volumes being related to the cube of the ratio of lengths.

2. Both small sphere A and large sphere B are spheres, and their volumes can be calculated based on the given radii. The formula for the volume of a sphere is V = 4/3 * π * r³.

3. The radius ratio of small sphere A to large sphere B is 3:12, thus the volume ratio can be derived as (3/12)³ = 1:64.

4. The volume of the large sphere B can be calculated using the formula as V_B = 4/3 * π * 12³.

5. V_B = 4/3 * π * 1728 = 2304 * π cm³.

6. If the volume ratio of small sphere A to large sphere B is 1:64, then the volume of small sphere A is 1/64 of the volume of the large sphere.

7. V_A = 2304 * π / 64 = 36 * π cm³.

8. Through the above reasoning, the final answer is 36π cm³.

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.

In the diagram of this problem, <note>small sphere A and large sphere B are both spheres</note>. <note>The radius of large sphere B, r_B, is 12 cm</note>, <note>the radius of small sphere A, r_A, is 3 cm</note>.

Formula for the Volume of a Sphere

In this problem, the volume of the large sphere <note>B的体积</note> V_B = (4/3)π <note>(12)</note>³ = 2304π cubic centimeters. The radius of the small sphere <note>A的半径</note> is <note>3 cm</note>, and its volume <note>体积</note> is V_A = (4/3)π <note>(3)</note>³ = 36π cubic centimeters. According to the similarity ratio calculation, V_A can also be verified by V_A = V_B * (<note>r_A</note>/<note>r_B</note>)³ = 2304π * (<note>3</note>/<note>12</note>)³ = 36π cubic centimeters.


Question: Find the surface area of the rectangular prism with length equal to 12 m.

Answer: 288 \mathrm{m}^2

1. Given the length l = 12 meters, width w = 6 meters, and height h = 4 meters of the cuboid.

2. According to the formula for the surface area of the cuboid S = 2 * (l * w + l * h + w * h), substitute l = 12 meters, w = 6 meters, and h = 4 meters.

3. Calculate the product of length and width: l * w = 12 meters * 6 meters = 72 square meters.

4. Calculate the product of length and height: l * h = 12 meters * 4 meters = 48 square meters.

5. Calculate the product of width and height: w * h = 6 meters * 4 meters = 24 square meters.

6. Sum the above three items: 72 square meters + 48 square meters + 24 square meters = 144 square meters.

7. Multiply the sum by 2: 2 * 144 square meters = 288 square meters.

8. Through the above reasoning, the final answer is 288 square meters.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>The three dimensions of a rectangular prism are: length l = 12 meters, width w = 6 meters, height h = 4 meters</note>. The six faces are rectangles formed by these lengths, widths, and heights.

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.

In the diagram of this problem, the surface area of the rectangular prism <note>S</note> is calculated using the formula <note>S = 2 * (l * w + l * h + w * h)</note>: <note>S = 2 * (12 meters * 6 meters + 12 meters * 4 meters + 6 meters * 4 meters)</note> = <note>2 * (72 square meters + 48 square meters + 24 square meters)</note> = <note>2 * 144 square meters</note> = <note>288 square meters</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, <note>the three faces of the rectangular prism are rectangles with lengths and widths, lengths and heights, and widths and heights</note>, so the areas of the rectangles are respectively:
Area of the rectangle with length and width = <note>12 meters * 6 meters = 72 square meters</note>,
Area of the rectangle with length and height = <note>12 meters * 4 meters = 48 square meters</note>,
Area of the rectangle with width and height = <note>6 meters * 4 meters = 24 square meters</note>.


Question:
Consider the rectangular prism shown with its height is 12m.

Write a fully expanded and simplified expression for the volume of the prism.

Answer: 12x^2+144x+384 \mathrm{m}^3

1. According to the formula for the volume of a cuboid, the volume of the rectangular prism is the length multiplied by the width multiplied by the height.

2. The length given in the problem is (x+4) meters, the width is (x+8) meters, and the height is 12 meters.

3. Therefore, the volume expression is: (x+4) * (x+8) * 12.

4. First, expand the first two factors (x+4) and (x+8):

5. (x+4)*(x+8)=x * x + x * 8 + 4 * x + 4 * 8.

6. Simplify to get: x² + 8x + 4x + 32.

7. Further simplify the result to: x² + 12x + 32.

8. Next, multiply the obtained polynomial by 12:

9. 12 * (x² + 12x + 32) = 12x² + 12 * 12x + 12 * 32.

10. Calculate each term to get: 12x² + 144x + 384.

11. After expanding and simplifying, the volume expression of the entire rectangular prism is obtained.

12. Through the above reasoning, the final answer is 12x² + 144x + 384.

Definition of Rectangular Prism

<note>Six faces are respectively the top and bottom faces, the front and back faces, and the left and right faces</note>. The given <note>height is 12 meters</note>, <note>length is (x+4) meters</note>, <note>width is (x+8) meters</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

<note>The length of the rectangular prism is (x+4) meters</note>, <note>the width is (x+8) meters</note>, <note>the height is 12 meters</note>. Therefore, the volume V of the rectangular prism is calculated as <note>V = (x+4) * (x+8) * 12</note>.


Question: Calculate the volume of the half cylinder with a diameter of 6 cm. Correct to one decimal place.

Answer: Volume \$=169.6 \mathrm{~cm}^{3}\$

1. Given that the diameter of the semicircular cylinder is 6 cm and its height is 12 cm, calculate the radius of the semicircle. According to the definition of radius, the radius is half of the diameter. Therefore, the radius r = 3 cm.

2. Calculate the area of the semicircle using the formula for the area of a circle A = (1/2)πr², where r is the radius. Substitute the value r = 3 cm, then A = (1/2)π(3 cm)² = (1/2)π * 9 cm² = 4.5π cm².

3. The volume of the semicircular cylinder is equal to the area of the semicircle multiplied by its height. Use the formula V = A * h, where A is the area of the semicircle and h is the height. Given the height h = 12 cm, substitute A = 4.5π cm², thus V = 4.5π cm² * 12 cm.

4. Calculate the volume to get V = 54π cm³.

5. Approximate π as 3.14, V ≈ 54 * 3.14 cm³ = 169.56 cm³.

6. Round the final result to one decimal place, getting 169.6 cm³.

Definition of Radius

In the figure of this problem, it is known that <note>the diameter of the semicircular cylinder is 6 cm</note>. According to the <note>definition of radius</note>, <note>the radius of the semicircle r = 3 cm</note>, which means <note>the distance from the center to the circumference is 3 cm</note>.

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.

In the figure of this problem, the semicircular cylinder's <note>diameter is 6 cm</note>, <note>the diameter is the line segment that passes through the center of the circle and has both endpoints on the circle</note>, it is the longest chord of the circle, with a length of 2 times the radius, i.e., <note>diameter = 2 * radius = 6 cm</note>, therefore <note>radius is 6 cm / 2 = 3 cm</note>.

Area Formula of a Circle

<note>In the figure of this problem, the radius of the semicircle is 3 cm</note>, according to the area formula of a circle, the area A of the circle is equal to the circumference π multiplied by the square of the radius 3, that is <note>A = π(3 cm)²</note>. Since the area of the semicircle is half of the full circle area, use the formula <note>A = (1/2)π(3 cm)²</note>, to calculate the area of the semicircle <note>4.5π cm²</note>.

Volume Formula of Prism

<note>The volume of a semicircular prism V = A * h</note>, where <note>A is the area of the semicircle</note>, <note>h is the height</note>. Given <note>height h = 12 cm</note>, substituting <note>semicircle area A = 4.5π cm²</note>, thus <note>V = 4.5π cm² * 12 cm = 54π cm³</note>.


Question: Find the volume of the prism shown. The height of the prism is 2cm.

Answer: \$299 \mathrm{~cm}^{3}\$

1. As shown in the figure, this is a wedge-shaped prism formed by translating the trapezoidal base around the side axis. The shape of the base of the prism is a trapezoid with the upper base being 7cm, the lower base being 16cm, and the distance between the two bases (i.e., the height of the trapezoid) being 2cm.

2. According to the trapezoid area formula, trapezoid area = (1/2) * (upper base + lower base) * height, we get: trapezoid area = (1/2) * (7cm + 16cm) * 2cm.

3. Calculate the area of the trapezoid: = (1/2) * (23cm) * 2cm = (1/2) * 46cm² = 23cm².

4. From the prism volume formula, base area * height, we get:

5. Base area = 23cm², height = 13cm, so prism volume = 23cm² * 13cm.

6. Finally, calculate the prism volume to get 23cm² * 13cm = 299cm³.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

In the figure of this problem, the base of the shape is a <note>trapezoid</note>, with its <note>upper base being 7cm</note>, <note>lower base being 16cm</note>, and <note>the distance between the two bases, which is the height of the trapezoid, is 2cm</note>. <note>The two non-parallel sides of the trapezoid are the two shorter slant sides in the figure</note>.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

<note>The base of the wedge-shaped prism is a trapezoid (the upper base is 7cm, the lower base is 16cm, and the height is 2cm)</note>, the height of the prism is <note>13cm</note>.

Trapezoid Area Formula

Given a trapezoid with the lengths of its two parallel sides as \(a\) and \(b\), and the height between these sides as \(h\), the area of the trapezoid is given by \(\frac{(a + b) \cdot h}{2}\).

<note>In the trapezoid base, the sides 7cm and 16cm are two parallel sides</note>, <note>the side 2cm is the height between them</note>, so the area of the trapezoid is (7cm + 16cm) * 2cm / 2.

Volume Formula of Prism

In the figure of this problem, use the volume formula of the prism to solve for the volume of the entire prism: Volume = <note>23cm² (base area)</note> * <note>13cm (height)</note> = 299cm³.


Question: The inner radius is 4 cm. Find the volume of concrete required to make the pipe, correct to two decimal places.

Answer: Volume \$=816.81 \mathrm{~cm}^{3}\$

1. The problem gives the inner diameter (D1) of the tubular structure as 4 cm, the outer diameter (D2) as 6 cm, and the length (L) as 13 cm. The task is to find the amount of cement concrete required, which is the volume of this tubular structure.

2. The formula needed for this problem is the cylinder volume formula. The volume of the pipe can be expressed as the difference between the volumes of two cylinders: V = πh(R2^2 − R1^2), where V is the volume, h is the height (i.e., length L), R2 is the outer radius of the cylinder, and R1 is the inner radius of the cylinder.

3. Using the pipe's length as the cylinder's height (13 cm), outer radius (R2 = 6 cm), and inner radius (R1 = 4 cm), we get: V = π × 13 cm × ((6 cm)^2 - (4 cm)^2).

4. Calculate the squares of the two radii: (6 cm)^2 = 36 cm², (4 cm)^2 = 16 cm².

5. According to the formula, the volume is: V = π × 13 × (36 cm² − 16 cm²) = π × 13 × 20 cm².

6. Specifically calculating, we get: V = π × 260 cm³ = 816.81 cm³ of concrete, with the result rounded to two decimal places.

7. Through the above reasoning, the required quantity is calculated to be 816.81 cm³.

Definition of Radius

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

In the diagram of this problem, <note>the inner radius of the cylinder R1 = 4 cm</note>, <note>the outer radius R2 = 6 cm</note>. <note>The radius is the line segment from the center of the circle to any point on the circle</note>, therefore <note>the distance of the line segment from the center of the circle to any point on the circle is the radius</note>.

Height of a Cylinder

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

In the figure of this problem, the height of the cylinder is <note>the length of the pipe, L = 13 cm</note>. This definition is used to determine the height that will be used to calculate the volume.

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>The volume of the pipeline can be expressed as the difference between the volumes of two cylinders</note>, which is <note>V = π × 13 cm × ((6 cm)^2 - (4 cm)^2)</note>. Calculating this yields <note>V = π × 13 cm × (36 cm² - 16 cm²) = π × 13 cm × 20 cm² = π × 260 cm³</note>, ≈ 816.81 cm³. This theorem is used in this problem to calculate the <note>total amount of cement concrete</note>.


Question: Find the volume of the cylinder shown, whose height is 8 cm, rounding your answer to two decimal places.

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

1. Given that the radius of the cylinder is 6 cm and the height is 8 cm.

2. The formula for the volume of the cylinder is V = π × r² × h, where r is the radius of the circle and h is the height of the cylinder.

3. Substitute the given r = 6 cm and h = 8 cm into the formula, and calculate V = π × (6 cm)² × 8 cm.

4. Further calculation gives V = π × 36 cm² × 8 cm = 288π cm³.

5. To calculate the approximate value, take π ≈ 3.14159, then V ≈ 3.14159 × 288 cm³.

6. Performing the multiplication gives V ≈ 904.77872 cm³.

7. Rounding to two decimal places as required, the volume of the cylinder is 904.78 cm³.

n. Through the above reasoning, the final answer is 904.78 cm³.

Cylinder

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

In the figure of this problem, <note>a cylinder consists of two parallel and identical circular bases and a lateral surface</note>. <note>The bases are two identical circles</note>, <note>with a radius of 6 cm</note>, and their centers lie on the same straight line. <note>The lateral surface is a rectangle, and when unfolded, its height equals the cylinder's height of 8 cm</note>, <note>and its width equals the circumference of the circle</note>.

Definition of Radius

<note>In the diagram of this problem, the base of the cylinder is a circle, the center is the center of the circle, and the distance from any point on the circumference to the center is 6 cm</note>, so this line segment is the radius of the circle.

Height of a Cylinder

<note>The height of the cylinder h is 8 cm</note>, that is, <note>the vertical distance between the two circular bases is 8 cm</note>.

Volume Formula of Cylinder

<note>Radius of the cylinder r=6 cm</note> and <note>height h=8 cm</note>, substituting these values into the formula yields the volume V = π × <note>(6 cm)²</note> × <note>8 cm</note> = 288π cm³.


Question: Find the surface area of the triangular prism shown. The height is 21 cm. The base is an isosceles triangle.

Answer: 768 \mathrm{cm}^2

1. Given that the base of the triangular prism is an isosceles triangle with side lengths of 10 cm and a base length of 12 cm, it can be seen that the height of the triangle base is 8 cm.

2. According to the triangle area formula, the area of the triangle is 1/2 * base * height. Therefore, the area of the base triangle is 1/2 * 12 cm * 8 cm = 48 cm².

3. The surface area of the triangular prism consists of two base triangles and three rectangular faces.

4. Among the three rectangular faces: one face comes from rotating along the base of the triangle, the area of this rectangle is 21 cm * 12 cm = 252 cm².

5. The other two faces come from rotating the two sides (10 cm each), resulting in two equal rectangles, so the area of each rectangle is 21 cm * 10 cm = 210 cm².

6. Therefore, the total area of the two side rectangles is 2 * 210 cm² = 420 cm².

7. Hence, the total surface area of the triangular prism = 2 * area of the base triangle + area of the rectangle along the base + area of the side rectangles.

8. The total surface area is 2 * 48 cm² + 252 cm² + 420 cm² = 768 cm².

9. Through the above reasoning, the final answer is 768 cm².

Area Formula of a Triangle

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

In the problem diagram, <note>the base of the triangle is 12 cm</note>, <note>the height is 8 cm</note>. According to the area formula of a triangle, the area of the triangle is equal to <note>the base multiplied by the height and then divided by 2</note>, that is, <note>Area = (12 cm * 8 cm) / 2 = 48 cm²</note>.

Definition of Isosceles Triangle

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

In the figure of this problem, <note>the lengths of the two legs of the triangle are both 10 cm</note>, <note>the length of the base is 12 cm</note>, therefore this triangle is an <note>isosceles triangle</note>.

Formula for the Area of a Rectangle

In the figure of this problem, the formula for the area of a rectangle is applied as follows:
1) The area of the rectangle formed along the <note>base of the triangle</note> is 21 cm * 12 cm = 252 cm²;
2) The area of the two rectangles formed along the <note>sides of the triangle</note>, each with an area of 21 cm * 10 cm = 210 cm², so the total area is 2 * 210 cm² = 420 cm².

Surface Area Formula for a Triangular Prism

The surface area of a triangular prism is equal to the sum of the areas of its two triangular bases plus the sum of the areas of its three rectangular lateral faces, that is, Surface Area = 2 * Base Area + Lateral Area.

<note>The total surface area of the triangular prism = 2 * the area of the base triangle + the area of the rectangle on the base edge + the area of the rectangular regions on the two sides</note>. According to the calculation, <note>2 * 48 cm² + 252 cm² + 420 cm² = 768 cm²</note>.


Question: As shown in the figure, the solid is constructed by a cone and semi-sphere. The height of the cone is 12cm. Find the volume of the composite figure shown, correct to two decimal places.

Answer: 169.65 \mathrm{cm}^3

1. Given that the radius of the hemisphere is 3 cm, the radius of the hemisphere = the radius of the base circle of the cone = 3 cm.

2. The formula for the volume of the hemisphere is: V_hemisphere = (2/3) * π * (radius)^3. Substituting the values: V_hemisphere = (2/3) * π * (3)^3 = 18π cm³.

3. For the cone, we know the radius of the base circle r = 3 cm and the height h = 12 cm. Using the formula for the volume of the cone: V_cone = (1/3) * π * r^2 * h. Substituting the values: V_cone = (1/3) * π * (3)^2 * 12 = 36π cm³.

4. The composite volume includes the volume of the cone and the volume of the hemisphere, so the total volume of the composite solid is: V_total = V_cone + V_hemisphere = 36π + 18π = 54π cm³.

5. Finally, calculate the total volume and round to two decimal places: V_total = 54π ≈ 169.65 cm³.

6. Through the above reasoning, the final answer is 169.65 cm³.

Definition of Hemisphere

A hemisphere is a three-dimensional geometric shape that constitutes half of a sphere, including half of the sphere's surface area and a circular base area.

<note>The radius of the hemisphere is 3 centimeters</note>, therefore <note>the diameter of the hemisphere is 6 centimeters</note>.

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 radius of the base circle of the cone is 3 cm</note>, <note>the height is 12 cm</note>.

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>The radius of the hemisphere is 3 cm</note>, so using the volume formula: <note>V_hemisphere = (2/3) * π * (3)^3 = 18π cm³</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>The radius of the cone is 3 cm</note>, <note>the height is 12 cm</note>, therefore, using the <note>volume formula</note> to calculate: V_cone = (1/3) * π * (3)^2 * 12 = 36π cm³.


Question:
All edges of the following cube have the same length.

Find the exact length of AG in simplest surd form.

Answer: \sqrt{147}

1. Given that the edge length of the cube is 7 cm, all edges of the cube are equal.

2. We need to find the distance AG from vertex A to vertex G. G is the diagonal vertex of the cube. We can calculate this using the space diagonal formula of the cube.

3. Two pairs of vertices of the cube form the space diagonal. These two vertices are on different top faces of the cube and are not on the same edge.

4. For the segment AG we need, we can use the space diagonal formula of the cube. The formula is: if the edge length of the cube is a, then the space diagonal length d is d = √(3a²).

5. Substitute the given edge length value a = 7, calculate AG = √147 = 7√3.

6. Take the product of 7 and √3 as the final answer.

7. Therefore, the distance AG between vertex A and vertex G calculated through the above reasoning steps is 7√3.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>In the diagram of this problem, each side of the cube is 7 cm long</note>. Specifically, <note>sides AB, BC, CD, DA, EF, FG, GH, HE are all 7 cm</note>.

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: \( d = a\sqrt{3} \)

<note>In the diagram of this problem, the side length of the cube is 7</note>,<note>the space diagonal d=√(3 * 7²)=√147=7√3</note>.


Question: Find the total surface area of the triangular prism shown. The height is 16cm.

Answer: 608 \mathrm{cm}^2

1. From the definition of a triangular prism, understand that the total surface area of the triangular prism is composed of the sum of the areas of two triangular bases and three rectangular lateral faces.

2. Let the triangle be ABC, B be the left vertex of the triangle base, and ABC be taken counterclockwise. Through point A, there is AH perpendicular to BC at H. Given that the length of the base of the triangle is 12 cm, according to the problem statement and the definition of a right triangle, triangles ABH and AHC are both right triangles. By the criterion of congruence for right triangles (hypotenuse and one leg), triangles ABH and AHC are congruent, so HC = 12/2. According to the Pythagorean theorem, which expresses the relationship between the two legs a, b and the hypotenuse c in a right triangle as a² + b² = c², we can find the height x of the triangle.

3. Applying the Pythagorean theorem (a² + b² = c²) to the given values, 10² = (12/2)² + x², which means 100 = 36 + x². Solving this, we get x² = 64, thus x = 8 cm. Therefore, the height of the triangle is 8 cm.

4. Using the area formula for a triangle, Area = 1/2 * base * height, calculate the area of the triangle = 1/2 * 12 cm * 8 cm = 48 cm².

5. The surface area of the triangular prism equals the sum of the areas of the two triangular bases plus the sum of the areas of the three rectangular lateral faces.

6. The sum of the areas of the two triangular bases: 2 * 48 cm² = 96 cm².

7. One lateral face is 12 cm (one side of the triangle base) * 16 cm (height of the prism) = 192 cm².

8. The other two rectangular lateral faces are 10 cm (the hypotenuse of the triangle) * 16 cm (height of the prism), so the area of the two identical lateral faces is: 2 * (10 cm * 16 cm) = 320 cm².

9. According to the surface area formula for a triangular prism, the total surface area of the three lateral faces and the two bases is 96 cm² + 192 cm² + 320 cm² = 608 cm².

10. Through the above reasoning, the final answer is 608 cm².

Area Formula of a Triangle

In the figure of this problem, <note>the base of the triangle is 12 cm</note>, <note>the height is 8 cm</note>. According to the <note>area formula of a triangle</note>, the area of the triangle is equal to <note>the base multiplied by the height and then divided by 2</note>, that is, Area = (12 cm * 8 cm) / 2 = 48 cm².

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 two legs of the right triangle are 6 cm (12/2)</note> and <note>x cm</note>, <note>the hypotenuse is 10 cm</note>. According to the Pythagorean Theorem, <note>the square of the hypotenuse is equal to the sum of the squares of the two legs</note>, that is, 10² = (12/2)² + x². Therefore, 100 = 36 + x², solving for x² = 64, hence <note>x = 8 cm</note>. Therefore, <note>the height of the triangle is 8 cm</note>.

In the figure of this problem, in the triangular prism, <note>the base is a triangle, and the sides are rectangles</note>. According to the surface area formula for a triangular prism, <note>the surface area of a triangular prism is equal to the sum of the areas of the two bases plus the sum of the areas of the three sides</note>. Specifically: 1. <note>The area of the base triangle is Base Area = 1/2 * 12 cm * 8 cm = 48 cm²</note>. 2. <note>The areas of the three side rectangles are Lateral Area1 = 12 cm * 16 cm = 192 cm², Lateral Area2 = 10 cm * 16 cm = 160 cm², and Lateral Area3 = 10 cm * 16 cm = 160 cm²</note>. Therefore, <note>the surface area of the triangular prism Surface Area = 2 * Base Area + Lateral Area1 + Lateral Area2 + Lateral Area3 = 2 * 48 cm² + 192 cm² + 160 cm² + 160 cm² = 608 cm²</note>.


Question: A pyramid has been removed from a rectangular prism, as shown in the figure. The height of the rectangular prism is 8cm. Find the volume of this composite solid.

Answer: 720 \mathrm{cm}^3

1. The volume formula for a rectangular prism is: length * width * height. The given length is 15 cm, width is 9 cm, and height is 8 cm.

2. According to the volume formula for a rectangular prism, the volume is: length * width * height = 15 cm * 9 cm * 8 cm = 1080 cubic centimeters.

3. The base area of the removed pyramid is the same as the base of the rectangular prism, i.e., the base area is length * width = 15 cm * 9 cm = 135 square centimeters.

4. The height of the removed pyramid is the height of the rectangular prism, which is 8 cm. The volume formula for a pyramid is 1/3 * base area * height.

5. According to the volume formula for a pyramid, the volume is: 1/3 * base area * height = 1/3 * 135 square centimeters * 8 cm = 360 cubic centimeters.

6. The volume of the composite solid can be obtained by subtracting the volume of the removed pyramid from the volume of the prism.

7. Therefore, the volume of the composite solid = volume of the rectangular prism - volume of the removed pyramid = 1080 cubic centimeters - 360 cubic centimeters = 720 cubic centimeters.

8. Through the above reasoning, the final answer is 720 cubic centimeters.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

In the figure of this problem, the base of the rectangular prism is a <note>15 cm * 9 cm rectangle</note>, and the top face is also a rectangle of the same size. The height is <note>8 cm, which is the distance between the two bases</note>. Each rectangle on the two bases in the figure has <note>four vertices marked with right-angle symbols, indicating that they are rectangles</note>.

Definition of Pyramid

A pyramid is a polyhedron formed by connecting a polygonal base with a point not in the same plane as the base (the apex) using line segments from each vertex of the polygon to the apex.

In the figure of this problem, the removed <note>pyramid base</note> is the same as the <note>rectangular prism base</note>, and the <note>base area is a 15 cm * 9 cm rectangle</note>. The <note>sides of the pyramid are perpendicular to the base</note> and the <note>vertex is located at the center point of the prism's base</note>. The figure shows that the four vertices of the base each form <note>four slant line segments with the vertex</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

The base area of the rectangular prism is <note>15 cm * 9 cm</note>, and the height is <note>8 cm</note>. Therefore, the volume of the rectangular prism is calculated as <note>15 cm * 9 cm * 8 cm = 1080 cm³</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: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

The original: <note>The base area of the removed pyramid is the same as the rectangular prism, which is 15 cm * 9 cm</note>, and <note>the height is also 8 cm</note>. Therefore, the volume of the pyramid is calculated as <note>1/3 * 135 cm² * 8 cm</note> = 360 cm³.


Question:
A rectangular prism has dimensions as labelled on the diagram. The height is 5.
Find the length of AG. Leave your answer in surd form.

Answer: \sqrt{122}

1. Given ABCD-EFGH is a cuboid, with length, width, and height being EF=9, GF=4, AE=5.

2. The problem requires finding the length of segment AG. Since AG is part of the cuboid's diagonal (space diagonal), and this diagonal passes through the length, width, and height of the cuboid, its formula is the cuboid's space diagonal formula.

3. In three-dimensional space, the formula for the cuboid's space diagonal is √(length² + height² + width²). Specifically, for this example, the length of AG is calculated as √(EF² + GF² + AE²).

4. Substitute the data into the calculation: EF=9, GF=4, AE=5, substitute into the formula to get AG=√(9² + 5² + 4²).

5. Perform the square operations on the data in the formula: 9²=81, 5²=25, 4²=16.

6. Substitute these values: AG=√(81 + 25 + 16).

7. Calculate the sum: 81 + 25 + 16 = 122.

8. The length of AG is: AG = √122.

9. Since the problem requires us to leave the answer in radical form, after the above reasoning, the final answer is √122.

Definition of Rectangular Prism

<note>This problem diagram is a rectangular prism ABCD-EFGH</note>, where the rectangular prism's <note>length is HG=9</note>, <note>width is GF=4</note>, <note>height is AE=5</note>.

Formula for the Space Diagonal of a Rectangular Prism

The length of the space diagonal of a rectangular prism is equal to the square root of the sum of the squares of the lengths of its three sides.

Original text: <note>The length, width, and height of the rectangular prism are a, b, and c respectively</note>, then the length of the space diagonal d can be calculated using the following formula: <note>d = √(a² + b² + c²) = x</note>.