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11-3 Surface of Pyramids 11-3 and 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 of • To find the surface area of a a pyramid Find the 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 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 • , lateral faces, , , 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 in which one (thebase ) can be any and be thought of as the limiting case the other faces (the lateral faces ) are 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 Lateral face If the base is a , 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 . altitude is the height h of the pyramid. Base edge Aregular pyramid is a pyramid whose Height Lesson Planning and base is a and whose lateral Resources faces are congruent isosceles triangles. h 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 of the areas of the congruent lateral faces. A 2s 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 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 Practice, Ch. 8 1 = 2pO The 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 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 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° to find 3 3, the of the " Quick Check 1 Find the surface area of a 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 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, 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. 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

Advanced Learners L4 English Language Learners ELL How much greater is the surface area of a Have students discuss how the formula for the surface than the surface area of a cone if the base radius of area of a 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 1 base is a circle. In a right cone, the altitude height lateral areas 2p and 22pr ? . is a perpendicular segment from the vertex 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 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 + ). 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 = 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. and B. much paper is in the cone? Round 2 2 = p(15)(25) + p(15) Substitute. 15 cm to the nearest tenth. about 2.5 ft = 375p + 225p Simplify. = 600p Resources • Daily Notetaking Guide 11-3 2 The surface area of the cone is 600p cm . L3 • Daily Notetaking Guide 11-3— Quick Check 3 The radius of the base of a cone is 22 m. Its slant height is 10 m. Find the surface Adapted Instruction L1 area in terms of p. 704π m2

By cutting a cone and laying it out flat, Closure you can see how the formula for lateral 1 / area of a cone L.A. 5 ? C ? O / How is finding the surface area Q 2 base R resembles that for the area of a of a pyramid like finding the A 5 1 surface area of a prism? How is triangle 2 bh . C base C Q R base it different? Sample: S.A. is the sum of the areas of the lateral Lesson 11-3 Surface Areas of Pyramids and Cones 619 faces and bases; the faces of a prism are rectangular and there are two bases, but the faces of a pyramid are triangular and there is one base.

619 3. Practice 4 EXAMPLE Real-World Connection

Chemistry The funnel is in the shape of a cone. How much 8 cm Assignment Guide filter paper do you need to the funnel?

1 The area covered is the lateral area of a cone that has a AB1-7, 14-20, 22, 27, 28, 7.5 cm 30-34, 36 diameter of 8 cm and a height of 7.5 cm. Use the formula for the lateral area of a cone. 2 8-13, 21, 23-26, 29, AB = O 35, 37, 38 L.A. pr 2 2 = pr a 1 b To find the slant height use the Pythagorean Theorem. C Challenge 39-43 A" B 2 2 = p(4) 4 1 7.5 Substitute. If d ≠ 8, then r ≠ 4. Careers Successful chemists A" B Test Prep 44-49 attend to detail, persevere, = 106.81415 Use a calculator. Mixed Review 50-54 and work independently. You need about 107 cm2 of filter paper to line the funnel. Homework Quick Check Quick Check 4 To check students’ understanding 4 Find the lateral area of a cone with radius 15 in. and height 20 in. 2 of key skills and concepts, go over 1178 in. Exercises 4, 12, 14, 26, 29.

Exercise 2 Have students explain EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. why the apothem equals 23 m. " Practice and Problem Solving

A Practice by Example Find the surface area of each pyramid to the nearest whole number. 179 in.2 Example 1 1.11 in. 2. 3. 7.2 in. (page 618) 138 m2 8 m

12 in. 8 in. 408 in.2 23 4 m GPS Guided Problem Solving L3 Enrichment L4 Example 2 Find the lateral area of each pyramid to the nearest whole number. 2 Reteaching L2 (page 618) 51 m 4.204 m2 5. 6. Adapted Practice L1 8 cm 6 m 6 m PracticeName Class Date L3 10 cm Practice 11-3 Inscribed For each diagram, indicate a pair of congruent inscribed angles. 5 3 cm

1.A 2.A 3. B B A 2 O 354 cm 4 m D 12 m C C B D E C D 4 m Find the value of each variable.

.5.90؇ 6 .4 ؇ y؇ x ؇ ؇ z x 110؇ 35؇ ؇ ؇ 7. Social Studies The original height of the pyramid built for Khafre, next 170 x y؇ x؇ 8.220؇ 9. 240؇ to the Great Pyramid, was about 471 ft. Each side of its square base was.7 O

62؇ 28؇ ؇ x؇ about 708 ft. What is the lateral area to the nearest foot of a pyramid with x ؇ z y؇ 2 ؇ those dimensions? 834,308 ft 10.y 11. 12. ؇ ؇ z؇ 54 100 ؇ ؇ x C ؇ y؇ 70 50؇ y ؇ z ؇ z؇ x؇ x Example 3 Find the surface area of each cone in terms of π. 2 Find each indicated measure for ᭪O. 0 119π cm a. AE 20؇ a. Є A 2 m ؇ 14. m A .13 10 B C B (page 619) 144π cm b. mЄC A b. mЄB 8. 9. 10. c. mЄBEC c. mЄC 100؇ d. Є d. Є ؇ m D m D O 120 O 10 cm D © Pearson Education, Inc. All rights reserved. D 8 ft 60؇ C 18 cm E 7 cm

6 ft 12 cm 33π ft2

620 Chapter 11 Surface Area and Volume

16. 228.1 in.2 4 in.

10 in. 620 Visual Learners Example 4 Find the lateral area of each cone to the nearest whole number. Exercise 17 Although students (page 620) 11. 12. 13. 26 in. 3 cm may use a formula, encourage 4.5 m them to sketch the figure before 22 in. solving the exercise. 4 cm 4 m 47 cm2 Exercise 21 Remind students that p is a constant and that they can 1044 in.2 31 m2 divide each side by the constant p to solve for ,. B Apply Your Skills 14. Writing Explain why the altitude PT in the P pyramid at the right must be shorter than Connection to Architecture each edge PA ,PB ,PC , and PD . See back of book. A B Exercise 23 Ask: Why might you Problem Solving Hint 15. Reasoning Suppose you could climb to the R want to know the area of a roof? In Exercise 14, explain top of the Great Pyramid in Egypt. Which T Sample: to estimate the cost of why PT is shorter than route would be shorter, a route along a lateral D C replacing roofing material PR, and then why PR is edge or along the altitude of a side? Which of shorter than PC . these routes is steeper? Explain your answers. See left. Exercise 30 Use this exercise to review Lessons 11-2 and 11-3. 16. A square pyramid has base edges 10 in. long and height 4 in. Sketch 15. Altitude; altitude; the Remind students not to include altitude is shorter the pyramid and find its surface area. Round your answer to the nearest tenth. See margin. the common base of the pyramid because it is one leg x 2 17. Algebra The lateral area of a pyramid with a square base is 240 ft2. Its base and prism. of a right k with the edges are 12 ft long. Find the height of the pyramid. 8 ft lateral edge as the hyp., and is steeper Diversity because it rises the Find the surface area to the nearest whole number. Exercise 43 You may want to 28 in.2 same vert. distance 18. 19. 20. review the formula for the area over less horiz. 6 cm of a sector of a circle. distance. 13 cm 4 in.

6 cm 4 in. 8 cm 2 478 cm2 62 cm 21. The lateral area of a cone is 48p in.2. The radius is 12 in. Find the slant height. 4 in. 22. Open-Ended Draw a square pyramid with a lateral area of 48 cm2. Label its dimensions. Then find its surface area. See margin. 23. Architecture The roof of a tower in a castle is shaped like a cone. The height of the roof is 30 ft and the radius of the base is 15 ft. What is the area of the roof? Round your answer to the nearest tenth. 1580.6 ft2

24. The hourglass shown at the right is made by connecting 6 in. two glass cones inside a glass cylinder. Which has more glass, the two cones or the cylinder? Explain. See back of book. 25. You can use the formula S.A. = ( + r) rp to find 8 in. the surface area of a cone. Explain why this formula works. Also, explain why you may prefer to use this Exercise 23 formula when finding surface area with a calculator. See back of book. 26. Find a formula for each of the following. 26a. / 5 S.A. 2 r pr a. the slant height of a cone in terms of the surface area and radius b. r = b. the radius of a cone in terms of the surface area and slant height 2p/ 1 p2/2 1 4p(S.A.) # 2p 27. Multiple Choice The roof of a tower is a square pyramid with base edges 10 ft long. The height of the pyramid is 6 ft. What is the approximate area of the roofing material needed to cover the roof? A 156 ft2 233 ft2 256 ft2 333 ft2

lesson quiz, PHSchool.com, Web Code: aua-1103 Lesson 11-3 Surface Areas of Pyramids and Cones 621

22. Answers may vary. Sample:

6 cm

4 cm 2 4 cm 64 cm 621 Find the surface area to the nearest whole number. 4. Assess & Reteach 45 m2 28. 29. 30. 3 m 5 ft 2 m PowerPoint GPS 6 ft Lesson Quiz 4 m 4 m 4 m 12 ft 1. Find the slant height of 3 m 2 a square pyramid with base 58 m 2 m 2 2 m edges 12 cm and altitude 471 ft 8 cm. 10 cm The length of a side of the base (s), slant height, height, lateral area, and surface 2. Find the lateral area of the area are measurements of a square pyramid. Given two of the measurements, find regular square pyramid below. the other three to the nearest tenth. s ≠ 12 m, L.A. ≠ 240 m2, S.A. ≠ 384 m2 ≠ 2 7 in. 31. L.A. 30 in. , 31. s = 3 in., S.A. = 39 in.2 See left. 32. h = 8 m, = 10 m h ≠ 4.8 in., , ≠ 5 in. 33. = 5 ft, L.A. = 20 ft2 34. L.A. = 118 cm2, S.A. = 182 cm2 s ≠ 2 ft, h ≠ 4.9 ft, S.A. ≠ 24 ft2 s ≠ 8 cm, , ≠ 7.4 cm, h ≠ 6.2 cm 35. A cone with radius 9 cm has the same surface area as a cylinder with radius 6 cm and height 18 cm. 37. cone with r ≠ 4 and 8 cm 2 What is the height of the cone to the nearest tenth? 4 in. 56 in. h ≠ 3; 36π 21.2 cm 38. cone with r ≠ 3 and 36. Find the surface area of the hexagonal pyramid 10 cm 2 3. Find the surface area of h ≠ 4; 24π at the right. about 613.5 cm 5 3 cm the pyramid whose base is a 39. cylinder with cone- Visualization The plane region is revolved regular hexagon. Round to the shaped hole; 60π y nearest whole number. completely about the given line to sweep out a 40. cylinder with cone- solid of revolution. Describe the solid. Then find its 2 shaped hole; 48π surface area in terms of π. 37–40. See left. 4242 O x 24 ft 37. about the y-axis 38. about the x-axis 2 C Challenge 39. about the line x = 4 40. about the line y = 3

The given figure fits inside a 10-cm . The figure’s base is in one face of the cube 73 ft and is as large as possible. The figure’s vertex is in the opposite face of the cube. 14 ft Draw a sketch and find the lateral and surface areas of the figure. 1517 ft2 41. a square pyramid 42. a cone 25π 5 cm2; 25π 5 ± 25π cm2 " " 4. Find the surface area of a cone 100 5 cm2; 100 5 ± 100 cm2 with radius 8 cm and slant 43. A" sector has been" cut out of the disk. The height 17 cm in terms of p. radii of the part that remains are taped 2 200π cm together, without overlapping, to form 5. The roof of a building is 129.6 the cone. The cone has a lateral area ? shaped like a cone with of 64p cm2. Find the measure of the 10 cm diameter 40 ft and height 20 central of the cut-out sector. ft. Find the surface area of the roof. Round to the nearest whole number. 1777 ft2 Test Prep Alternative Assessment Multiple Choice 44. To the nearest whole number, what is the surface area of a cone with diameter 27 m and slant height 19 m? A Pair students, and have one draw A. 1378 m2 B. 1951 m2 C. 2757 m2 D. 3902 m2 a pyramid and label its dimensions and the other draw a cone and 45. To the nearest whole number, what is the surface area of a cone with label its dimensions. Each student radius 14 cm and slant height 18 cm? J should calculate the surface area F. 448 cm2 G. 836 cm2 H. 1012 cm2 J. 1407 cm2 on a separate sheet of paper. Then have students exchange 622 Chapter 11 Surface Area and Volume drawings, calculate the surface areas, and compare their answers for both figures.

622 46. To the nearest whole number, what is the surface area of a square pyramid Test Prep with each side of the base 30 yd and slant height 42 yd? C A. 900 yd2 B. 2520 yd2 C. 3420 yd2 D. 3600 yd2 Resources 47. A cylinder and a cone each have height 1 and radius 3 . How does the For additional practice with a ! variety of test item formats: cylinder’s lateral area x compare with the cone’s lateral area y? F Standardized Test Prep, p. 657 F. x = y G. x = 2y H. x . 2y J. x , 2y • • Test-Taking Strategies, p. 652 Short Response 48. A square pyramid is 8 m on each side. Its surface area is 240 m2. What is its • Test-Taking Strategies with slant height? Show your work and explain your reasoning. See margin. Transparencies Extended Response 49. The lateral area of a cone is twice the area of its base. a. What is its slant height in terms of the radius r? Show your work. b. What is the lateral area to the nearest tenth if the radius is 6 centimeters? Show your work. a–b. See margin.

Use this Checkpoint Quiz to check students’ understanding of the MixedMixed ReviewReview skills and concepts of Lessons 11-1 through 11-3. Lesson 11-2 50. How much cardboard do you need to make a closed box that is 4 ft by 5 ft by 2 ft? Resources 76 ft2 51. How much posterboard do you need to make a cylinder, open at each end, with Grab & Go 1 • Checkpoint Quiz 1 height 9 in. and diameter 4 2 in.? Round your answer to the nearest square inch. 127 in.2 Lesson 10-2 52. The area of a is 714 cm2. One is 42 cm long. Find the length of the other diagonal. 34 cm 53. A with area 195 in.2 has a 15-in. diagonal. How long is the other diagonal? 26 in. Lesson 8-5 54. A TV camera views a tall building 400 m away with a 358 angle of elevation to the top. How tall is the building if the camera lens is 160 cm off the ground? Checkpoint Quiz 1 about 281.7 m 8. Answers may vary. Sample: Checkpoint Quiz 1 Lessons 11-1 through 11-3

Draw a net for each figure. Label the net with its dimensions. 2–3. See back of book. See left. 1. 1. 2. 3. 40 m 4 cm 12 in. 11 cm 60 m 11 cm 60 m 8π cm 4 cm 6.3 in. 9. 4 in.

2 4 cm 4. Find the surface area of the cylinder in Exercise 1. 120π cm 5. Find the surface area of the prism in Exercise 2. 297.6 in.2 9600 m2 6. Find the surface area of the pyramid in Exercise 3. 10. 7. A cone has a radius of 8 in. and a height of 10 in. Find the lateral area of the cone. Leave your answer in terms of p. 104π in.2 8. Open-Ended Draw a net for a regular hexagonal prism. See margin.

Draw a cube. Shade the cube to show each cross section. 9–10. See margin. 9. a 10. a

Lesson 11-3 Surface Areas of Pyramids and Cones 623

48. [2] Use the formula for [1] correct formula with L.A. ≠ 226.2 cm2 [1] correct lateral area surface area of a one computational error without any work ≠ 1 ± [3] one computational pyramid: S.A. 2 p, ≠ shown B. Subst. the surface 49. [4]a. L.A. 2B error , ≠ 2 area, perimeter, and πr 2πr , ≠ [2] correct methods in base: 240 ≠ 16, ± 64. 2r ≠ (a) and (b) but Solve: , ≠ 11. b. L.A. πr, L.A. ≠ π(6)(12) incorrect relationship 623 between , and r