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13.2 Volume of Pyramids and Cones

13.2 Volume of Pyramids and Cones. Vinh Lam and Zach Mann. Objectives. Find volume of pyramids. Find volume of cones. Things to Know. For this lesson you will need to know how to use Trigonometry to solve for angles, as well as rules for 30-60-90 and 45-45-90 triangles. . Volume of Pyramids.

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13.2 Volume of Pyramids and Cones

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  1. 13.2 Volume of Pyramids and Cones Vinh Lam and Zach Mann

  2. Objectives • Find volume of pyramids. • Find volume of cones.

  3. Things to Know • For this lesson you will need to know how to use Trigonometry to solve for angles, as well as rules for 30-60-90 and 45-45-90 triangles.

  4. Volume of Pyramids • If a pyramid has a volume of V cubic units, a height of h units, and a base with an area of B square units, then... V = 1/3Bh

  5. Example #1 • Find the volume of a square pyramid if it has a slant height of 34 in., a base edge of 32 in., and a height of 30 in. Round your answer to the nearest tenth. V = 1/3Bh

  6. Example 1 Solution • V = 1/3(32²)(30) Plug in • V = 1/3(30,720) Multiply • V = 10,240 in.³ Multiply by 1/3

  7. Your Turn! Find the volume of a pentagonal pyramid with a base edge of 4 and a height of 10. Round your answer to the nearest tenth. 4

  8. Example #2 Solution • First, find the apothem of the pentagon to find the area of the base. The apothem is x. • Tan36° = 2/x Trig. • xTan36° = 2 Multiply • x = 2/Tan36° Divide • x = 2.752763841 Solve

  9. Example # 2 Solution (Cont.) • Since you now have the apothem, plug it in to find the area of the pentagon. (1/2Pa) • B = ½(20)(2.752763841) Equation • B = 27.52763841 Solve • Now plug the area of the base in to find the area of the pyramid. • V = 1/3(27.52763841)(10) Equation • V = 91.8 Solve

  10. Volume of Cones • If a right circular cone has a volume of V cubic units, a height of h units, and the base has a radius of r units, then... V = 1/3πr²h

  11. Example #1 • Find the volume of a cone if it has a radius of 15 and a height of 22. Round your answer to the nearest tenth. V = 1/3πr²h 22 15

  12. Example #1 Solution • V = 1/3π(15²)(22)Plug in • V = 1/3π(4,950)Multiply • V = 1,650πMultiply • V = 5,183.6 Solve

  13. Your Turn! • Find the volume of a cone with a radius of 12. Round your answer to the nearest tenth. 45° 12

  14. Example #2 Solution • Using the rules of 45-45-90 triangles, you know the legs are equal, so the height is 12. • V = 1/3π(12²)(12) Plug in • V = 1/3π(1,728) Multiply • V = 576π Multiply • V = 1,809.6 Solve

  15. Volume of Oblique Cones • The formula for the volume of an oblique cone is the same as that of a right cone, so… V = 1/3πr²h

  16. Example #1 • Find the volume of an oblique cone if it has a diameter of 30 feet and a height of 25 feet. Round your answer to the nearest tenth. V = 1/3πr²h

  17. Example #1 Solution • We know the diameter is 30, so the radius is 15 feet, so your equation should read.. • V = 1/3π(15²)(25) Equation • V = 1/3π(5,625) Multiply • V = 1875π Multiply • V = 5890.5 Solve

  18. Your Turn! • Find the volume of an oblique cone with a height of 14 and a radius of 6. Round your answer to the nearest tenth. 14 6

  19. Example #2 Solution • V = 1/3πr²h Equation • V = 1/3π(6²)(14) Plug in • V = 1/3π(504) Multiply • V = 168π Multiply • V = 527.8 Solve

  20. Homework Assignment SECANT!!!! Pg. 699 #8-19 all, 25, 26

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