Find the area of each polygon or circle. 1. Trapezoid, bases 12 ft and 18 ft, height 3 ft ANSWER 324 ft2 2. Circle, diameter 8.2 in. ANSWER 7.27 in.2 3. Regular hexagon, side length 14 cm ANSWER 509.22 cm2

Vocabulary volume – of a solid is the number of cubic units contained in its interior. Volume Congruence Postulate 28 – If two polyhedra are congruent, they have the same volume. Volume Addition Postulate 29 – The volume of a solid is the sum of the volumes of all its nonoverlapping parts.

Vocabulary Volume of a Cube Postulate 27 – V = s3 Volume of a Prism Theorem 11.6 – V = Bh Volume of a Cylinder Theorem 11.7 – V = Bh = πr 2h where B = base area, h = height, r = radius, s = side

Vocabulary Cavalieri’s Principle – If two solids have the same height and the same cross- sectional area at every level, then they have the same volume.

EXAMPLE 1 Find the number of unit cubes 3- D PUZZLE Find the volume of the puzzle piece in cubic units.

EXAMPLE 1 Find the number of unit cubes SOLUTION To find the volume, find the number of unit cubes it contains. Separate the piece into three rectangular boxes as follows: The base is 7 units by 2 units. So, it contains 7 . 2, or 14 unit cubes. The upper left box is 2 units by 2 units. So, it contains 2 . 2, or 4 unit cubes. The upper right box is 1 unit by 2 units. So, it contains 1 . 2, or 2 unit cubes. By the Volume Addition Postulate, the total volume of the puzzle piece is 14 + 4 + 2 = 20 cubic units.

EXAMPLE 2 Find volumes of prisms and cylinders Find the volume of the solid. a. Right trapezoidal prism SOLUTION (3)(6 + 14) a. The area of a base is 1 2 = 30cm2 and h = 5 cm. Use h = 5 cm to find the volume. V = Bh = 30(5) = 150cm3

EXAMPLE 2 Find volumes of prisms and cylinders Find the volume of the solid. b. Right cylinder SOLUTION b. The area of the base is π 92, or 81πft2. Use h = 6 ft to find the volume. V = Bh = 81π(6) = 486π ≈ 1526.81 ft3

The volume of the cube is 90 cubic inches. Find the value of x. EXAMPLE 3 Use volume of a prism ALGEBRA The volume of the cube is 90 cubic inches. Find the value of x. SOLUTION A side length of the cube is x inches. V = x3 Formula for volume of a cube 90 in3 = x3 Substitute for V. 4.48 in ≈ x Find the cube root.

GUIDED PRACTICE for Example 1,2,and 3 1. Find the volume of the puzzle piece shown in cubic units. The volume of the puzzle cube is 7 units3 ANSWER 2. Find the volume of a square prism that has a base edge length of 5 feet and a height of 12 feet. The volume of a square prism is 300 ft3 ANSWER

GUIDED PRACTICE for Example 1,2,and 3 3. The volume of a right cylinder is 684π cubic inches and the height is 18 inches. Find the radius. The radius of a right cylinder is in 38 ANSWER

Find the volume of an oblique cylinder EXAMPLE 4 Find the volume of an oblique cylinder Find the volume of the oblique cylinder. SOLUTION Cavalieri’s Principle allows you to use Theorem 12.7 to find the volume of the oblique cylinder. V = π r2h Formula for volume of a cylinder = π(42)(7) Substitute known values. = 112π Simplify. ≈ 351.86 Use a calculator. The volume of the oblique cylinder is about 351.86 cm3. ANSWER

EXAMPLE 5 Solve a real-world problem PLANTER The planter is made up of 13 beams. In centimeters, suppose the dimensions of each beam are 30 by 30 by 90. Find its volume. SOLUTION The area of the base B can be found by subtracting the area of the small rectangles from the area of the large rectangle. B = Area of large rectangle – 4 Area of small rectangle

Solve a real-world problem EXAMPLE 5 B = Area of large rectangle – 4 Area of small rectangle = 90 510 – 4(30 90) = 35,100 cm2 Use the formula for the volume of a prism. V = Bh Formula for volume of a prism. = 35,100(30) Substitute. = 1,053,000 cm3 Simplify. The volume of the planter is 1,053,000 cm3, or 1.053 m3.

GUIDED PRACTICE for Example 4 and 5 4. Find the volume of the oblique prism shown below. The volume of the oblique prism is 180m3 ANSWER 5. Find the volume of the solid shown below. ANSWER 205.81 ft3

Daily Homework Quiz 1. Find the volume of the solid at the right. ANSWER 5437.5 in.3 2. Find the volume of a right triangular prism with height 32 in., base height 12 in., and base length 18 in. ANSWER 3456 in.3 3. Find the volume of a right cylinder with height 30 ft and diameter 14 ft. ANSWER 4618.14 in.3

Daily Homework Quiz 4. A cylindrical beaker with diameter 10 in. and height 12 in. is filled with water that is then poured into a rectangular pan that is 14 in. by 9 in. by 3 in. What is the volume of each solid? Would the water overflow the pan? If so, what is the height of the water in the beaker after the pan is filled? ANSWER cylinder: 942.48 in.3; pan: 378 in.3; Yes, it would overflow the pan. After the pan is filled , the height of the water in the beaker will be about 7.2 in.