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Published byDiana Sutton Modified over 5 years ago

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

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**Vocabulary vertex of a pyramid regular pyramid**

slant height of a regular pyramid altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a 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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**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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**Example 1B: Finding Lateral Area and Surface Area of Pyramids**

Find the lateral area and surface area of the regular pyramid. Step 1 Find the base perimeter and apothem. The base perimeter is 6(10) = 60 in. The apothem is , so the base area is

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Example 1B Continued Find the lateral area and surface area of the regular pyramid. Step 2 Find the lateral area. Lateral area of a regular pyramid Substitute 60 for P and 16 for ℓ.

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Example 1B Continued Find the lateral area and surface area of the regular pyramid. Step 3 Find the surface area. Surface area of a regular pyramid Substitute for B.

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Check It Out! Example 1 Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 1 Find the base perimeter and apothem. The base perimeter is 3(6) = 18 ft. The apothem is so the base area is

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**Check It Out! Example 1 Continued**

Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 2 Find the lateral area. Lateral area of a regular pyramid Substitute 18 for P and 10 for ℓ.

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**Check It Out! Example 1 Continued**

Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. Step 3 Find the surface area. Surface area of a regular pyramid Substitute for B.

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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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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.

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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.

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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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Check It Out! Example 2 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 3: Exploring Effects of Changing Dimensions**

The base edge length and slant height of the regular hexagonal pyramid are both divided by 5. Describe the effect on the surface area.

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**base edge length and slant height divided by 5: original dimensions:**

Example 3 Continued base edge length and slant height divided by 5: original dimensions: S = Pℓ + B 12 S = Pℓ + B 12

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**base edge length and slant height divided by 5: original dimensions:**

Example 3 Continued base edge length and slant height divided by 5: original dimensions: Notice that If the base edge length and slant height are divided by 5, the surface area is divided by 52, or 25.

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Check It Out! Example 3 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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**Check It Out! Example 3 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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**Example 4: Finding Surface Area of Composite Three-Dimensional Figures**

Find the surface area of the composite figure. Left-hand cone: The lateral area of the cone is L = rl = (6)(12) = 72 in2. Right-hand cone: Using the Pythagorean Theorem, l = 10 in. The lateral area of the cone is L = rl = (6)(10) = 60 in2.

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Example 4 Continued Find the surface area of the composite figure. Composite figure: S = (left cone lateral area) + (right cone lateral area) = 60 in2 + 72 in2 = 132 in2

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Check It Out! Example 4 Find the surface area of the composite figure. Surface Area of Cube without the top side: S = 4wh + B S = 4(2)(2) + (2)(2) = 20 yd2

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**Check It Out! Example 4 Continued**

Surface Area of Pyramid without base: Surface Area of Composite: Surface of Composite = SA of Cube + SA of Pyramid

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