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**Surface Area of 10-5 Pyramids and Cones Warm Up Lesson Presentation**

Lesson Quiz Holt Geometry

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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 = 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

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

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**The vertex of a pyramid is the point opposite the base of the pyramid**

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 Pl Surface Area of a regular pyramid with lateral area L and base area B is ________________________ S = L + B or S = ½ Pl + 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 = 142 = 196 cm2

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**The vertex of a cone is the point opposite the base**

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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**L = rl S = L + B or S = rl + r2 Lateral Area of a right cone**

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 = rl Surface Area of a right cone with lateral area L and base area B is _______________________ S = L + B or S = rl + r2

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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 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. 10

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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 = 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.

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**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 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.

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**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.

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**multiplied by two-thirds:**

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

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