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10-5

Surface Area of Pyramids and Cones. 10-5. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Warm Up Find the missing side length of each right triangle with legs a and b and hypotenuse c . 1. a = 7, b = 24 2. c = 15, a = 9 3. b = 40, c = 41 4. a = 5, b = 5

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10-5

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  1. Surface Area of Pyramids and Cones 10-5 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up Find the missing side length of each right triangle with legs a and b and hypotenuse c. 1.a = 7, b = 24 2.c = 15, a = 9 3.b = 40, c = 41 4.a = 5, b = 5 5.a = 4, c = 8 c = 25 b = 12 a = 9

  3. Objectives Learn and apply the formula for the surface area of a pyramid and cone.

  4. The vertex of a pyramidis the point opposite the base of the pyramid. The base of a regular pyramidis a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramidis the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramidis the perpendicular segment from the vertex to the plane of the base.

  5. 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.

  6. Lateral Area of a regular pyramid with perimeter, P and slant height l is _________ L = ½ Pl Surface Area of a regular pyramid with lateral area L and base area B is ________________________ S = L + B or S = ½ Pl + B

  7. Example 1A: Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary. Lateral area of a regular pyramid P = 4(14) = 56 cm Surface area of a regular pyramid B = 142 = 196 cm2

  8. The vertex of a coneis the point opposite the base. The axis of a coneis the segment with endpoints at the vertex and the center of the base. The axis of a right coneis perpendicular to the base. The axis of an oblique coneis not perpendicular to the base.

  9. The slant height of a right coneis the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a coneis a perpendicular segment from the vertex of the cone to the plane of the base. Lateral Area of a right cone with radius, r and slant height l is _______ L = rl Surface Area of a right cone with lateral area L and base area B is _______________________ S = L + B or S = rl + r2

  10. Example 2A: Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of a right cone with radius 9 cm and slant height 5 cm. L = rℓ Lateral area of a cone = (9)(5) = 45 cm2 Substitute 9 for r and 5 for ℓ. S = rℓ + r2 Surface area of a cone = 45 + (9)2 = 126 cm2 Substitute 5 for ℓ and 9 for r.

  11. Example 2B: Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of the cone. Use the Pythagorean Theorem to find ℓ. L = rℓ Lateral area of a right cone = (8)(17) = 136 in2 Substitute 8 for r and 17 for ℓ. S = rℓ+ r2 Surface area of a cone = 136 + (8)2 = 200 in2 Substitute 8 for r and 17 for ℓ.

  12. ℓ Example 2C: Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of the right cone. Use the Pythagorean Theorem to find ℓ. L = rℓ Lateral area of a right cone = (8)(10) = 80 cm2 Substitute 8 for r and 10 for ℓ. S = rℓ + r2 Surface area of a cone = 80 + (8)2 = 144 cm2 Substitute 8 for r and 10 for ℓ.

  13. Example 3A: Exploring Effects of Changing Dimensions The base edge length and slant height of the regular square pyramid are both multiplied by . Describe the effect on the surface area.

  14. 8 ft 12 12 10 ft S = Pℓ + B S = Pℓ + B By multiplying the dimensions by two-thirds, the surface area was multiplied by . Example 3A Continued 8 ft original dimensions: multiplied by two-thirds: = 585 cm2 = 260 cm2

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