Volume of Prisms and Cylinders Warm Up Lesson Presentation Lesson Quiz

Warm-Up 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

Example 1 Find the volume of the puzzle piece in cubic units. 3-D PUZZLE

Example 1 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 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. V = Bh = 30(5) = 150cm3

Example 2 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 The volume of the cube is 90 cubic inches. Find the value of x. ALGEBRA 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 1. Find the volume of the puzzle piece shown in cubic units. 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. 300 ft3 ANSWER

Guided Practice 3. The volume of a right cylinder is 684π cubic inches and the height is 18 inches. Find the radius. in. 38 ANSWER

Find the volume of the oblique cylinder. Example 4 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.

Example 5 The sculpture is made up of 13 beams. In centimeters, suppose the dimensions of each beam are 30 by 30 by 90. Find its volume. SCULPTURE 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 = 90 510 – 4(30 90) = 35,100 cm2

Use the formula for the volume of a prism. Example 5 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 sculpture is 1,053,000 cm3, or 1.053 m3.

Guided Practice 4. Find the volume of the oblique prism shown below. 180 m3 ANSWER

Guided Practice 5. Find the volume of the solid shown below. about 205.81 ft3 ANSWER

Lesson 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

Lesson Quiz 3. Find the volume of a right cylinder with height 30 ft and diameter 14 ft. ANSWER 4618.14 in.3

Lesson 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.