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

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

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

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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.

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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.

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

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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 = 14 2 = 196 cm 2

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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.

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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. Lateral Area of a right cone with radius, r and slant height l is _______ L = r l Surface Area of a right cone with lateral area L and base area B is _______________________ S = L + B or S = r l + r 2

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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 cm 2 Substitute 9 for r and 5 for ℓ. S = rℓ + r 2 Surface area of a cone = 45 + (9) 2 = 126 cm 2 Substitute 5 for ℓ and 9 for r.

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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 = r ℓ = (8)(17) = 136 in 2 Lateral area of a right cone Substitute 8 for r and 17 for ℓ. S = r ℓ + r 2 Surface area of a cone = 136 + (8) 2 = 200 in 2 Substitute 8 for r and 17 for ℓ.

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

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

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Example 3A Continued original dimensions:multiplied by two-thirds: By multiplying the dimensions by two-thirds, the surface area was multiplied by. 8 ft 10 ft S = Pℓ + B 1212 = 260 cm 2 S = Pℓ + B 1212 = 585 cm 2

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