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Surface Areas of Pyramids and Cones 617 11-3 Surface Areas of Pyramids 11-3 and Cones 11-3 1. Plan What You’ll Learn Check Skills You’ll Need GO for Help Lesson 8-1 Objectives 1 To find the surface area of • To find the surface area of a a pyramid pyramid Find the length of the hypotenuse in simplest radical form. 313 cm 2 To find the surface area of 1. 2.130 m 3. " • To find the surface area of a " a cone cone 8 in. 9 m 13 cm Examples . And Why 233 in. 1 Finding Surface Area of " 13 in. 7 m To find the lateral area of the 12 cm a Pyramid Great Pyramid of Egypt, as in Example 2 2 Real-World Connection New Vocabulary • pyramid • base, lateral faces, vertex, altitude, height, 3 Finding Surface Area of slant height, lateral area, surface area (of a pyramid) a Cone • regular pyramid • cone • base, altitude, vertex, height, 4 Real-World Connection slant height, lateral area, surface area (of a cone) • right cone Math Background 1 Finding Surface Area of a Pyramid The fact that the lateral area of both a pyramid and cone equals 1 2pl is not coincidental. A cone can Apyramid is a polyhedron in which one face (thebase ) can be any polygon and be thought of as the limiting case the other faces (the lateral faces) are triangles that meet at a common vertex of a regular pyramid, just as a (called the vertex of the pyramid). Vertex circle can be thought of as the You can name a pyramid by the shape of Lateral limiting case of a regular n-gon. Vocabulary Tip its base. The altitude of a pyramid is the edge Lateral face If the base is a hexagon, perpendicular segment from the vertex to Altitude the pyramid is a the plane of the base. The length of the More Math Background: p. 596C Base hexagonal pyramid. altitude is the height h of the pyramid. Base edge Aregular pyramid is a pyramid whose Height Lesson Planning and base is a regular polygon and whose lateral Resources faces are congruent isosceles triangles. h ᐉ Slant O height The slant height is the length of the See p. 596E for a list of the altitude of a lateral face of the pyramid. resources that support this lesson. In this book, you can assume that a pyramid is regular unless stated otherwise. PowerPoint Bell Ringer Practice The lateral area of a pyramid is the sum 1 ᐉ of the areas of the congruent lateral faces. A 2s ᐉ Check Skills You’ll Need You can find a formula for the lateral area For intervention, direct students to: of a pyramid by looking at its net. s ᐉ ᐉ Simplest Radical Form 1 1 ssBase L.A. = 4 sO The area of each lateral face is sO. Q2 R 2 Lesson 8-1: Example 2 1 Commutative and Associative s Extra Skills, Word Problems, Proof = (4s)O 2 Properties of Multiplication ᐉ Practice, Ch. 8 1 = 2pO The perimeter p of the base is 4s. Lesson 11-3 Surface Areas of Pyramids and Cones 617 Special Needs L1 Below Level L2 Have students create pyramids from nets or using Have students construct and label a net for a straws or pipe cleaners. For each pyramid, have hexagonal pyramid to use with Example 1. students measure the slant height and the length of an edge of the base. learning style: tactile learning style: visual 617 2. Teach To find the surface area of a pyramid, add the area of its base to its lateral area. Key Concepts Theorem 11-3 Lateral and Surface Areas of a Regular Pyramid Guided Instruction The lateral area of a regular pyramid is half the product of the perimeter of the base and the slant height. 1 ᐉ Error Prevention! L.A. = 2 pO Students may confuse the height The surface area of a regular pyramid is the sum of the h and slant height ᐉ of a pyra- lateral area and the area of the base. B mid. Suggest that students use S.A. = L.A. + B the word slant as a cue that the height being measured is along a slanted triangular face. 1 EXAMPLE Finding Surface Area of a Pyramid Tactile Learners Have models of pyramids and Find the surface area of the hexagonal pyramid at the left. = + cones for students to touch as 9 in. S.A. L.A. B Use the formula for surface area. you discuss the definitions and 1 1 ͙ = pO + ap Substitute the formulas for L.A. and B. theorems in this lesson. 3 3 2 2 1 1 = (36)(9) + 3 3 (36) Substitute. 2 2A ! B 1 EXAMPLE Teaching Tip 6 in. < 255.53074 Use a calculator. Make sure that students remember The surface area of the pyramid is about 256 in.2. how to use a 30°-60°-90° triangle to find 3 3, the apothem of the " Quick Check 1 Find the surface area of a square pyramid with base edges 5 m and slant height 3 m. hexagonal base. 55 m2 PowerPoint Sometimes the slant height of a pyramid is not given. You must calculate it before Additional Examples you can find the lateral or surface area. 1 Find the surface area of a 2 EXAMPLE Real-World Connection square pyramid with base edges 7.5 ft and slant height 12 ft. Social Studies The Great Pyramid at Giza, Egypt, A 2 481 ft 236.25 ft pictured at the left, was built about 2580 B.C. as a final resting place for Pharaoh Khufu. At the time it was built, ᐉ 2 Find the lateral area of the its height was about 481 ft. Each edge of the square base C hexagonal pyramid below. Round was about 756 ft long. What was the lateral area of the B 756 ft to the nearest whole number. Great Pyramid? A The legs of right #ABC are the height of the pyramid and the apothem of the base. The height of the pyramid was 481 ft. ᐉ 756 481 ft The apothem of the base was2 , or 378 ft. You can use the Pythagorean Theorem to find the slant height O. BC 20 m 378 ft Real-World Connection 1 L.A. = 2 pO Use the formula for lateral area. Today, most casing stones ͙ 1 2 2 4 3 m (used to smooth the sides) and = 2(4s) a 1 b Substitute the formulas for p and O. 8 m " some of the top stones are 1 2 2 = (4 ? 756) 378 1 481 Substitute. about 508 m2 gone from this pyramid. 2 " < 924974.57 Use a calculator. The lateral area of the Great Pyramid was about 925,000 ft2. Quick Check 2 Find the surface area of the Great Pyramid to the nearest square foot. 1,496,511 ft2 618 Chapter 11 Surface Area and Volume Advanced Learners L4 English Language Learners ELL How much greater is the surface area of a cylinder Have students discuss how the formula for the surface than the surface area of a cone if the base radius of area of a prism and the formula for the surface area each is 6 in. and their heights are 10 in.? 108π in. of a regular pyramid are alike and different. 618 learning style: verbal learning style: verbal Guided Instruction 12 Finding Surface Area of a Cone Math Tip A cone is “pointed” like a pyramid, but its Slant Have students compare the 1 ᐉ 1 ᐉ base is a circle. In a right cone, the altitude height lateral areas 2p and 22pr ? . is a perpendicular segment from the vertex ᐉ Altitude Ask: How are the formulas to the center of the base. The height h is the h similar? The perimeter of a length of the altitude. The slant height O is r pyramid’s base is like the the distance from the vertex to a point on the circumference of a cone’s base. edge of the base. Base Connection 3 EXAMPLE to Algebra 1 As with a pyramid, the lateral area is 2 the perimeter (circumference) of the base Challenge students to justify times the slant height. The formulas for the lateral area and surface area of a cone why the formula for the surface are similar to those for a pyramid. area of a cone can be written S.A. = pr(r + ᐉ). Key Concepts Theorem 11-4 Lateral and Surface Areas of a Cone The lateral area of a right cone is half the product of the PowerPoint circumference of the base and the slant height. Additional Examples ᐉ = 1 ? ? O = O L.A. 2 2pr , or L.A. pr 3 Find the surface area of the The surface area of a right cone is the sum of the lateral r cone in terms of p. B area and the area of the base. S.A. = L.A. + B 13 in. 12 in. In this book, you can assume that a cone is a right cone unless stated or pictured otherwise. 5 in. 3 EXAMPLE Finding Surface Area of a Cone 90π in.2 Find the surface area of the cone in terms of p. 4 Leandre uses paper cones to cover her plants in the early S.A. = L.A. + B Use the surface area formula. 25 cm spring. The diameter of each cone Substitute the formulas = prO + pr2 is 1 ft, and its height is 1.5 ft. How for L.A.
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