Volume of Pyramids and Cones Warm Up Lesson Presentation Lesson Quiz

Warm-Up Find the area of each polygon or circle. 1. Regular hexagon, side length 9 in. ANSWER 210.44 in.2 2. Circle, radius 15 m ANSWER 706.86 m2 3. Right triangle, hypotenuse 12 cm, side length 6 cm ANSWER 31.18 cm

Warm-Up 4. Solve x = 15 3. (2 3 )2 3 12 ANSWER 15

Example 1 Find the volume of the solid. a. V = Bh 1 3 = ( 4 6)(9) 1 3 2 = 36 m3

Example 1 Find the volume of the solid. b. V = Bh 1 3 = (π 2.22)(4.5) 1 3 = 7.26π ≈ 22.81 cm3

Originally, the side length of the base was about 215 m. Example 2 Originally, the pyramid had height 144 meters and volume 2,226,450 cubic meters. Find the side length of the square base. ALGEBRA SOLUTION V = bh Write formula. 2,226,450 = (x2)(144) Substitute. 6,679,350 = 144x2 Multiply each side by 3. 46,384 ≈ x2 Divide each side by 144. 215 ≈ x Find the positive square root. Originally, the side length of the base was about 215 m.

Guided Practice Find the volume of the solid. Round your answer to two decimal places, if necessary. 1. Hexagonal pyramid 152.42 yd3 ANSWER

Guided Practice Find the volume of the solid. Round your answer to two decimal places, if necessary. 2. Right cone 163.49 m3 ANSWER

Guided Practice 3. The volume of a right cone is 1350π cubic meters and the radius is 18 meters. Find the height of the cone. 12.5 m ANSWER

Find the volume of the right cone. Example 3 Find the volume of the right cone. SOLUTION To find the radius r of the base, use trigonometry. tan 65° = opp. Adj. Write ratio. tan 65° = 16 r Substitute. r = tan 65° 16 ≈ 7.46 Solve for r.

Example 3 Use the formula for the volume of a cone. V = (π r 2)h ≈ π(7.462)(16) ≈ 932.45 ft3 3 1

Find the volume of the solid shown. Example 4 Find the volume of the solid shown. SOLUTION = s3 + Bh 1 3 Write formulas. = 63 + (6)2 6 1 3 Substitute. = 216 + 72 Simplify. = 288 Add. The volume of the solid is 288 cubic meters.

Example 5 You are using the funnel shown to measure the coarseness of a particular type of sand. It takes 2.8 seconds for the sand to empty out of the funnel. Find the flow rate of the sand in milliliters per second. (1 mL = 1 cm3) SCIENCE SOLUTION STEP 1 Find the volume of the funnel using the formula for the volume of a cone. V = (πr2)h 1 3 = π(42)(6) 1 3 ≈ 101 cm3 = 101 mL

Example 5 STEP 2 Divide the volume of the funnel by the time it takes the sand to empty out of the funnel. 101 mL 2.8 s ≈ 36.07 mL/s The flow rate of the sand is about 36.07 milliliters per second.

Guided Practice 4. Find the volume of the cone at the right. Round your answer to two decimal places. 143.86 in3 ANSWER

Guided Practice 5. A right cylinder with radius 3 centimeters and height 10 centimeters has a right cone on top of it with the same base and height 5 centimeters. Find the volume of the solid. Round your answer to two decimal places. 330 cm2 ANSWER

Guided Practice WHAT IF? In Example 5, suppose a different type of sand is used that takes 3.2 seconds to empty out of the funnel. Find its flow rate. 6. 101ml 3.25 = 31.56 ml/s ANSWER

Lesson Quiz Find the volume of each solid. 1. ANSWER 96 in.3

Lesson Quiz Find the volume of each solid. 2. ANSWER 3619.11 mm3

Lesson Quiz 3. Find the volume of the cube after the cone is removed. ANSWER 232.11 mm3

Lesson Quiz 4. Find the volume of a right cone with height 32 in. and radius r in. is 2144.66 in.3. Find r. ANSWER 8.0 in.