8-8 Surface Area of Pyramids and Cones Warm Up Problem of the Day Lesson Presentation Course 3

Warm Up 1. A rectangular prism is 0.6 m by 0.4 m by 1.0 m. What is the surface area? 2. A cylindrical can has a diameter of 14 cm and a height of 20 cm. What is the surface area to the nearest tenth? Use 3.14 for . 2.48 m2 1186.9 cm2

Problem of the Day Sandy is building a model of a pyramid with a hexagonal base. If she uses a toothpick for each edge, how many toothpicks will she need? 12

Learn to find the surface area of pyramids and cones.

Vocabulary slant height regular pyramid right cone

The slant height of a pyramid or cone is measured along its lateral surface. Regular Pyramid Right cone The base of a regular pyramid is a regular polygon, and the lateral faces are all congruent. In a right cone, a line perpendicular to the base through the tip of the cone passes through the center of the base.

Additional Example 1: Finding Surface Area Find the surface area of each figure to the nearest tenth. Use 3.14 for p. 1 2 A. S = B + Pl = (2.4 • 2.4) + (9.6)(3) 1 2 = 20.16 ft2 B. S = pr2 + prl = p(32) + p(3)(6) = 27p  84.8 cm2

Check It Out: Example 1 Find the surface area of each figure to the nearest tenth. Use 3.14 for p. 5 m 1 2 A. S = B + Pl = (3 • 3) + (12)(5) 1 2 3 m = 39 m2 3 m B. S = pr2 + prl 18 ft = p(72) + p(7)(18) 7 ft = 175p  549.5 ft2

Additional Example 2: Exploring the Effects of Changing Dimensions A cone has diameter 8 in. and slant height 3 in. Explain whether tripling the slant height would have the same effect on the surface area as tripling the radius. They would not have the same effect. Tripling the radius would increase the surface area more than tripling the slant height.

Triple the Slant Height Check It Out: Example 2 A cone has diameter 9 in. and a slant height 2 in. Explain whether tripling the slant height would have the same effect on the surface area as tripling the radius. Original Dimensions Triple the Slant Height Triple the Radius S = pr2 + prl S = pr2 + pr(3l) S = p(3r)2 + p(3r)l = p(4.5)2 + p(4.5)(2) = p(4.5)2 + p(4.5)(6) = p(13.5)2 + p(13.5)(2) = 29.25p in2  91.8 in2 = 47.25p in2  148.4 in2 = 209.25p in2  657.0 in2 They would not have the same effect. Tripling the radius would increase the surface area more than tripling the height.

Additional Example 3: Application The upper portion of an hourglass is approximately an inverted cone with the given dimensions. What is the lateral surface area of the upper portion of the hourglass? a2 + b2 = l2 Pythagorean Theorem 102 + 242 = l2 l = 26 24 mm L = prl Lateral surface area = p(10)(26)  816.8 mm2

Check It Out: Example 3 A large road construction cone is almost a full cone. With the given dimensions, what is the lateral surface area of the cone? a2 + b2 = l2 Pythagorean Theorem 92 + 122 = l2 12 in. 9 in. l = 15 L = prl Lateral surface area = p(4)(15)  188.4 in2

Lesson Quiz: Part I Find the surface area of each figure to the nearest tenth. Use 3.14 for p. 1. the triangular pyramid 2. the cone 6.2 m2 175.8 in2

Lesson Quiz: Part II 3. Tell whether doubling the dimensions of a cone will double the surface area. It will more than double the surface area because you square the radius to find the area of the base.