1. Volume of Pyramids and Cones Section 9.5

2. Objectives: Find the volumes of pyramids and cones.

3. Key Vocabulary Pyramid Cone Volume

4. A pyramid is a polyhedron in which base, that is a polygon, and lateral faces which are triangles. base vertex Pyramid

5. Volume of a Pyramid The volume of a prism is equal to Bh, where B is the area of the base, and h is the height. From the figure at the left, it is clear that the volume of the pyramid with the same base area B and the same height h must be less than the volume of the prism.

6. Volume of a Pyramid The volume of the pyramid is one third the volume of the prism.

7. Volume of Regular Pyramid Symbols: B = area of base h = height Words: 1/3(area base)(height)

8. Find the Volume of a Pyramid Example 1 Find the volume of the pyramid. The volume is 56 cubic meters. ANSWER The volume is 40 cubic feet. ANSWER a. b. = 40 Simplify. = 56 SOLUTION Write the formula for volume. V = Bh 3 1 a. V = Bh 3 1 b. Substitute. (5 · 4)(6) 3 1 = 3 1 = · 7 · 6 (8) 2 1

9. Checkpoint Find the Volume of a Pyramid Find the volume of the pyramid. ANSWER 84 in. 3 ANSWER 120 ft 3 ANSWER 180 cm 3 1. 2. 3.

10. vertex r height base Cone A cone has a circular base and a vertex that is not in the same plane as the base. The height of a cone is the perpendicular distance between the vertex and the base

11. Volume of a Cone The volume of a cylinder is equal to π r 2 h, where πr 2 is the area of the base, and h is the height. From the figure at the left, it is clear that the volume of the cone with the same base area and the same height must be less than the volume of the cylinder. The volume of the cone is one third the volume of the cylinder.

12. Volume of a Cone r = radius h = height

13. Volume of a cone

14. Find the Volume of a Cone Example 2 The volume is about 804 cubic centimeters. ANSWER Find the volume of the cone. Round your answer to the nearest whole number. SOLUTION The radius of the cone is r = 8 cm. The height of the cone is h = 12 cm. V =  r 2 h 3 1 Write the formula for volume of a cone. ≈ 804 Multiply. Substitute 8 for r and 12 for h. =  (8 2 )(12) 3 1

15. Find the Volume of a Cone Example 3 What is the volume of the cone shown at the right? SOLUTION You are given the slant height of the cone. You need to find the height of the cone before you can find the volume. Find the height.1. (leg) 2 + (leg) 2 = (hypotenuse) 2 Use the Pythagorean Theorem. 3 2 + h 2 = 5 2 Substitute. 9 + h 2 = 25 Simplify.

16. Find the Volume of a Cone Example 3 h 2 = 25 – 9 Subtract 9 from each side. h 2 = 16 Simplify. h = 4h = 4 3 1 V =  r 2 h Write the formula for volume. =  (3 2 )(4) 3 1 Substitute 3 for r and 4 for h. ≈ 38 Multiply. ANSWER The volume is about 38 cubic inches. Find the volume.2. Take the positive square root. h =h = 16

17. Your Turn: Find the volume of the cone. Round your answer to the nearest whole number. 1. 2. 3. ANSWER 236 in. 3 ANSWER 183 ft 3 ANSWER 2513 m 3

18. Your Turn: Find the volume of the cone. Round your answer to the nearest whole number. 4. Find the volume of a cone with a height of 6 inches and a diameter of 8 inches. ANSWER 101 in. 3 5.Find the volume of a cone with a slant height of 17 feet and a diameter of 16 feet. ANSWER 1005 ft 3

19. Assignment Pg. 513 – 516: #1 – 4 all, 5 – 51 odd