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Geometry Surface Area of Pyramids and Cones

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  1. GeometrySurface Area of Pyramids and Cones CONFIDENTIAL

  2. Warm Up Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth. 10 cm 3. 1. 2. 8 cm 2 cm 17 in. 15 in. 3 cm 15 cm 10 in. CONFIDENTIAL

  3. Surface Area of Pyramids and Cones The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. Vertices Lateral faces Altitude Bases Slant height regular pyramid nonregular pyramid Next page: CONFIDENTIAL

  4. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. P = 4s l l s s s s s s s s CONFIDENTIAL

  5. Lateral and surface Area of a regular pyramid The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl. The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B. l CONFIDENTIAL

  6. Finding Lateral and surface Area of Pyramids Find the lateral area and surface area of each pyramid. a.) a regular square pyramid with base edge length 5 in. and slant height 9in. L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in. S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in 2 2 2 2 Next page: CONFIDENTIAL

  7. b.)Find the lateral area and surface area of each regular pyramid. find the base perimeter and apothem. The base perimeter is 6(4) = 24 m. The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 7 m 2 Step 1 4 m Find the lateral area. L = ½ Pl Lateral area of a regular pyramid =½ (24)(7) = 84 m Substitute 24 for P and 7 for l. Step 2 Find the surface area. S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. Step 3 2 CONFIDENTIAL

  8. Now you try! 1) Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10ft. CONFIDENTIAL

  9. The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. Vertices Lateral surfaces Slant height axis axis Base oblique cone right cone Next page: CONFIDENTIAL

  10. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. CONFIDENTIAL

  11. Lateral and Surface Area of a right cone The lateral area of a right cone with radius r and slant height l is L = rl. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r l 2 r l r CONFIDENTIAL

  12. Finding Lateral Area and Surface Area of right cones Find the lateral area and surface area of each cone. Give your answers in terms of . A) A right cone with radius 2 m and slant height 3 m. Next page: CONFIDENTIAL

  13. B) l 12 ft Step 1 Use the Pythagorean Theorem to find l. 2 l = 5 + 12 = 13 ft 2 5 ft Step 2 Find the lateral area and surface area. CONFIDENTIAL

  14. Now you try! 2) Find the lateral area and surface area of the right cone. 16 ft 6 ft CONFIDENTIAL

  15. Exploring Effects of Changing Dimensions 5 cm The radius and slant height of the right cone tripled. Describe the effect on the surface area. 3 cm Radius and slant height tripled: Original dimensions: CONFIDENTIAL

  16. Now you try! 3) The base edge length and slant height of the regular square pyramid are both multiplied by 2/3 . Describe the effect on the surface area. 12 ft 15 ft CONFIDENTIAL

  17. Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. 28 cm 90 cm 45 cm Next page: CONFIDENTIAL

  18. 28 cm 90 cm 45 cm CONFIDENTIAL

  19. Now you try! 4) Find the surface area of the composite figure. 2 yd 2 yd 2 yd CONFIDENTIAL

  20. Electronics Application Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in. The diameter of the cone is 2(7.5) = 15 in. 10 in. 2 Substitute 10 for l, the slant height of the cone and the radius of the large circle. Solve it 2 CONFIDENTIAL

  21. Now you try! 5) If the radius of the large circle were 12 in., what would be the radius of the cone? CONFIDENTIAL

  22. Now some problems for you to practice ! CONFIDENTIAL

  23. Assessment 1) Describe the endpoints of an axis of a cone. CONFIDENTIAL

  24. 2) Find the lateral area and surface area of each regular pyramid. b. a. 12 cm 15 ft 16 ft 8 cm 16 ft CONFIDENTIAL

  25. 3) Find the lateral area and surface of each right cone. Given your answer in terms of . b. a. 24 in. 22 m. 25 in. 14 m. CONFIDENTIAL

  26. 4) Describe the effect of each change on the surface area of the given figure. b. The dimension are tripled. a. The dimensions are cut in half. 10 in. 15 cm. 6 in. 9 cm. 6 in. CONFIDENTIAL

  27. 5) Find the surface area of each composite figure. b. a. 15 ft. 26 m. 8 ft. 12 m. 18 ft. 32 m. 15 m. CONFIDENTIAL

  28. 6) Anna is making a birthday hat from a pattern that is ¾ of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain. 6 in. CONFIDENTIAL

  29. Let’s review CONFIDENTIAL

  30. Surface Area of Pyramids and Cones The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. Vertices Lateral faces Altitude Bases Slant height regular pyramid nonregular pyramid Next page: CONFIDENTIAL

  31. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. P = 4s l l s s s s s s s s CONFIDENTIAL

  32. Lateral and surface Area of a regular pyramid The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl. The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B. l CONFIDENTIAL

  33. Finding Lateral and surface Area of Pyramids Find the lateral area and surface area of each pyramid. a.) a regular square pyramid with base edge length 5 in. and slant height 9in. L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in. S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in 2 2 2 2 Next page: CONFIDENTIAL

  34. b.)Find the lateral area and surface area of each regular pyramid. find the base perimeter and apothem. The base perimeter is 6(4) = 24 m. The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 7 m 2 Step 1 4 m Find the lateral area. L = ½ Pl Lateral area of a regular pyramid =½ (24)(7) = 84 m Substitute 24 for P and 7 for l. Step 2 Find the surface area. S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. Step 3 2 CONFIDENTIAL

  35. The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. Vertices Lateral surfaces Slant height axis axis Base oblique cone right cone Next page: CONFIDENTIAL

  36. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. CONFIDENTIAL

  37. Lateral and Surface Area of a right cone The lateral area of a right cone with radius r and slant height l is L = rl. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r l 2 r l r CONFIDENTIAL

  38. Finding Lateral Area and Surface Area of right cones Find the lateral area and surface area of each cone. Give your answers in terms of . A) A right cone with radius 2 m and slant height 3 m. Next page: CONFIDENTIAL

  39. B) l 12 ft Step 1 Use the Pythagorean Theorem to find l. 2 l = 5 + 12 = 13 ft 2 5 ft Step 2 Find the lateral area and surface area. CONFIDENTIAL

  40. Exploring Effects of Changing Dimensions 5 cm The radius and slant height of the right cone tripled. Describe the effect on the surface area. 3 cm Radius and slant height tripled: Original dimensions: CONFIDENTIAL

  41. Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. 28 cm 90 cm 45 cm Next page: CONFIDENTIAL

  42. 28 cm 90 cm 45 cm CONFIDENTIAL

  43. Electronics Application Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in. The diameter of the cone is 2(7.5) = 15 in. 10 in. 2 Substitute 10 for l, the slant height of the cone and the radius of the large circle. Solve it 2 CONFIDENTIAL

  44. You did a great job today! CONFIDENTIAL