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Section 12.3 Surface Areas of Pyramids and Cones

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Lateral Area and Surface Area of Pyramids The lateral faces of a pyramid intersect at a common point called the vertex. Two lateral faces intersect at a lateral edge. A lateral face and the base intersect at a base edge. The altitude is the segment from the vertex perpendicular to the base. A regular pyramid has a base that is a regular polygon and the altitude has an endpoint at the center of the base. All the lateral edges are congruent and all the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height, ℓ, of a pyramid.

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The lateral area L of a regular pentagonal pyramid is the sum of the areas of all its congruent triangular faces as shown in the net at the right.

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Example 1: Find the lateral area of the square pyramid to the nearest tenth. Answer: The lateral area is 25 square centimeters.

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The surface area of a pyramid is the sum of the lateral area and the area of the base.

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Example 2: Find the surface are of the square pyramid to the nearest tenth. Find the slant height c 2 = a 2 + b 2 Pythagorean Theorem ℓ 2 = 6 2 + 4 2 a = 6, b = 4, and c = ℓ Simplify Find the perimeter and area of the base. P= 4 ● 8 or 32 mA = 8 2 or 64 m 2

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Answer: The surface area of the pyramid is about 179.4 square meters. Find the surface area of the pyramid. S= Pℓ + B Surface area of a regular pyramid __ 1 2 1 2 = (32) + 64 P = 32, ℓ =, B = 64 ≈ 179.4 Use a calculator.

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Example 3: Find the surface area of the regular pyramid. Round to the nearest hundredth. Find the perimeter of the base. P = 6 ● 10.4 or 62.4 cm Find the length of the apothem and the area of the base. A central angle of the hexagon is 60°, so the angle formed in the triangle is 30°.

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So, the area of the base is approximately 280.8 cm 2.

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Find the surface area of the pyramid. S= Pℓ + B Surface area of a regular pyramid __ 1 2 1 2 = (62.4)(15) + 280.8 P = 62.4, ℓ = 15, B = 280.8 ≈ 748.8 Use a calculator. Answer: The surface area of the pyramid is about 748.8 cm 2.

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Lateral Area and Surface Area of Cones Recall that a cone has a circular base and a vertex. The axis of a cone is the segment with endpoints at the vertex and the center of the base. If the axis is also the altitude, then the cone is a right cone. If the axis is not the altitude, then the cone is an oblique cone.

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Example 4: Find the surface area of the cone. Round to the nearest tenth. S= rℓ + r 2 Surface area of a cone = (1.4)(3.2) + (1.4) 2 r = 1.4 and ℓ = 3.2 ≈20.2 Answer: The surface area of the cone is about 20.2 square centimeters.

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Example 5: A sugar cone has an altitude of 8 inches and a diameter of 2.5 inches. Find the lateral area of the sugar cone. If the cone has a diameter of 2.5 inches then the radius = 0.5(2.5) = 1.5 in. Use the altitude and the radius to find the slant height with the Pythagorean Theorem. Find the slant height ℓ. ℓ 2 =8 2 + 1.25 2 Pythagorean Theorem ℓ 2 ≈65.56Simplify. ℓ≈8.1Take the square root of each side. Find the lateral area L. L= rℓ Lateral area of a cone ≈ (1.25)(8.1)r = 1.25 and ℓ ≈ 8.1 ≈31.8 Answer: The lateral area of the sugar cone is about 31.8 square inches.

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