**Volume**

The volume of a geometric figure is a measure of its capacity. Volume is measured in cubic units. Cubic units are abbreviated using 3 for the exponent:

- 1 cubic inch = 1 in.
^{3} - 1 cubic foot = 1 ft
^{3} - 1 cubic yard = 1 yd
^{3}

The basic geometric solids are the rectangular solid, the cube, the cylinder, the sphere, the right circular cone, and the pyramid. These figures are shown in Fig. 9-16.

**Volume of a Rectangular Solid**

*The volume of a rectangular solid can be found by using the formula V = lwh, where l = the length, w = the width, and h = the height* .

**Example 1**

Find the volume of the rectangular solid shown in Fig. 9-17 .

**Solution 1**

**Volume of a Cube**

*The volume of a cube can be found by using the formula V = s ^{3} , where s is the length of the side* .

**Example 2**

Find the volume of the cube shown in Fig. 9-18 .

**Solution 2**

**Volume of a Cylinder**

*The volume of a cylinder can be found by using the formula V = πr ^{2} h, where r is the radius of the base and h is the height* .

**Example 3**

Find the volume of the cylinder shown in Figure 9-19 . Use π = 3.14.

**Solution 3**

**Volume of a Sphere**

*The volume of a sphere can be found by using the formula * , where r is the radius of the sphere .

**Example 4**

Find the volume of the sphere shown in Figure 9-20. Use π = 3.14.

**Solution 4**

**Volume of a Right Circular Cone**

*The volume of a right circular cone can be found by using the formula * , where r is the radius of the base and h is the height of the cone .

**Example 5**

Find the volume of the cone shown in Fig. 9-21 . Use π = 3.14.

**Solution 5**

**Volume of a Pyramid**

*The volume of a pyramid can be found by using the formula * , where B is the area of the base and h is the height of the pyramid. If the base is a square, use B = s ^{2} . If the base is a rectangle, use B = lw .

**Example 6**

Find the volume of the pyramid shown in Fig. 9-22 .

**Solution 6**

In this case, the base is a square, so the area of the base is B = s ^{2} .

**Converting to Cubic Measurements**

Sometimes it is necessary to convert from cubic yards to cubic feet, cubic inches to cubic feet, etc. The following information will help you to do this. To change:

- cubic feet to cubic inches, multiply by 1728;
- cubic inches to cubic feet, divide by 1728;
- cubic yards to cubic feet, multiply by 27;
- cubic feet to cubic yards, divide by 27.

**Examples**

**Example 7**

Change 18 cubic yards to cubic feet.

**Solution 7**

18 × 27 = 486 cubic feet

**Example 8**

Change 15,552 cubic inches to cubic feet.

**Solution 8**

15,552 ÷ 1728 = 9 cubic feet

**Volume Practice Problems**

**Practice**

1. Find the volume of a rectangular solid if it is 8 inches long, 6 inches wide, and 10 inches high.

2. Find the volume of a cube if the side is 13 feet.

3. Find the volume of a cylinder if the radius is 3 inches and its height is 7 inches. Use π = 3.14.

4. Find the volume of a sphere if its radius is 9 inches. Use π = 3.14.

5. Find the volume of a cone if its radius is 6 feet and its height is 8 feet. Use π = 3.14.

6. Find the volume of a pyramid if its height is 2 yards and its base is a rectangle whose length is 3 yards and whose width is 2.5 yards.

**Answers**

1. 480 in. ^{3}

2. 2197 ft ^{3}

3. 197.82 in. ^{3}

4. 3052.08 in. ^{3}

5. 301.44 ft ^{3}

6. 5 yd ^{3}

Practice problems for these concepts can be found at: Informal Geometry Practice Test.

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