11-2 11-2 Surface Areas of Prisms and Cylinders 1. Plan Objectives 1 2 To find the surface area of a prism To find the surface area of a cylinder Examples 1 2 3 4 Finding Surface Area of a Prism Using Formulas to Find Surface Area Finding Surface Area of a Cylinder Real-World Connection What You’ll Learn Check Skills You’ll Need • To find the surface area of a Find the area of each net. prism • To find the surface area of a GO for Help 1. 96 cm2 40π cm2 2. cylinder 4 cm 4 cm . . . And Why 3. Lessons 1-9 and 10-3 36"3 m2 8 cm 4 cm To find the area covered by a drum on a roller used in road construction, as in Example 4 6m 4 cm New Vocabulary • prism • bases, lateral faces, altitude, height, lateral area, surface area (of a prism) • right prism • oblique prism • cylinder • bases, altitude, height, lateral area, surface area (of a cylinder) • right cylinder • oblique cylinder Math Background This lesson uses the area formulas from Chapter 10 and nets to develop formulas for the lateral and surface areas of prisms and cylinders. Nets especially simplify finding these areas for nonright prisms, such as parallelepipeds. The key idea is that the lateral faces of any prism are parallelograms. 1 Finding Surface Area of a Prism A prism is a polyhedron with exactly two congruent, parallel faces, called bases. Other faces are lateral faces. You name a prism by the shape of its bases. Bases Lateral edges Bases Lateral faces More Math Background: p. 596C An altitude of a prism is a perpendicular segment that joins the planes of the bases. The height h of the prism is the length of an altitude. Lesson Planning and Resources A prism may either be right or oblique. See p. 596E for a list of the resources that support this lesson. Real-World Pentagonal prism Triangular prism h Connection h A triangular prism breaks white light into rainbow colors. h right prisms oblique prism PowerPoint In a right prism the lateral faces are rectangles and a lateral edge is an altitude. In this book you may assume that a prism is a right prism unless stated or pictured otherwise. Bell Ringer Practice Check Skills You’ll Need For intervention, direct students to: The lateral area of a prism is the sum of the areas of the lateral faces. The surface area is the sum of the lateral area and the area of the two bases. Finding Area Lesson 1-9: Examples 4–6 Extra Skills, Word Problems, Proof Practice, Ch. 1 Areas of Regular Polygons Lesson 10-3: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 10 608 Chapter 11 Surface Area and Volume Special Needs Below Level L1 Draw examples of prisms on the board and have students identify the bases, lateral faces, and altitudes. Point out that in a rectangular prism, any pair of opposite faces could be considered the bases. 608 learning style: visual L2 Making nets of rectangular prisms may help students understand and remember Theorem 11-1. learning style: tactile 1 2. Teach Finding Surface Area of a Prism EXAMPLE Use a net to ﬁnd the surface area of the prism at the left. 4 3 Surface Area = Lateral Area + area of bases = sum of areas of lateral faces + area of bases 5 = (5 ? 4 + 5 ? 3 + 5 ? 4 + 5 ? 3) + 2(3)(4) 5 cm 4 3 Some students may think that a right prism must contain a right angle in its base. Point out that the right angle is formed by the lateral faces meeting the base. The surface area of the prism is 94 cm2. Quick Check 1 Use a net to ﬁnd the surface area of the triangular prism. See left. 5 cm 5 cm 1. 216 cm2 6 cm 2 12 cm You can ﬁnd formulas for lateral and surface areas by looking at a net for a prism. 5 cm Perimeter of base abcd a b c d 12 cm c d Base a b c d a h b h c Guided Instruction Error Prevention! 4 = 94 4 cm 3 3 = 70 + 24 3 cm 4 EXAMPLE Alternative Method Have students calculate the sum of the areas of the lateral faces to help them understand why the formula L.A. = ph makes sense. d PowerPoint h Additional Examples 6 cm Perimeter Lateral Area ph Height Area of d Base Lateral Area a base c Surface Area L.A. 2B 1 Use a net to find the surface area of the cube. You can use the formulas with any right prism. 2 EXAMPLE Using Formulas to Find Surface Area 11 in. Multiple Choice What is the surface area of the prism? B A 1 4 5 A A B B C By the Pythagorean Theorem, the hypotenuse of the triangular base is 5 cm. E D C 84 cm2 726 in.2 96 cm2 2 Find the surface area of a 10-cm high right prism with triangular bases having 18-cm edges. Round to the nearest whole number. 821 cm2 E D C B 78 cm2 E D C B A 3 C B A 2 72 cm2 E D C D D E E Test-Taking Tip A question could ask for either surface area or lateral area of a solid. Read the question carefully. L.A. = ph = 12 ? 6 3 cm 4 cm Use the formula for lateral area. p ≠ 3 ± 4 ± 5 ≠ 12 cm 6 cm = 72 The lateral area of the prism is 72 cm 2. Now use the formula for surface area. S.A. = L.A. + 2B = 72 + 2(6) = 84 B ≠ 12 (3 ? 4) ≠ 6 cm2 The surface area of the prism is 84 cm2. Choice C is the answer. Quick Check 2 Use formulas to ﬁnd the lateral area and surface area of the prism. 432 m2; about 619 m2 12 m 6m Lesson 11-2 Surface Areas of Prisms and Cylinders Advanced Learners 609 English Language Learners ELL L4 After Example 2, ask students to write a formula for the surface area of a rectangular prism with edges of length <, w, and h. S.A. ≠ 2(h< ± hw ± <w) learning style: verbal To help students with the new vocabulary, have them make and display a poster of a prism and a cylinder, labeling their parts. Clarify that the base of a prism is a face but the base of a polygon is a side. learning style: visual 609 Guided Instruction Key Concepts Theorem 11-1 Tactile Learners Lateral and Surface Areas of a Prism The lateral area of a right prism is the product of the perimeter of the base and the height. Have students tape the sides of a sheet of paper together (without overlapping) to form a cylinder. Ask: What is the lateral area of the cylinder? the area of the paper L.A. = ph p is the perimeter of a base. h B is the area of a base. The surface area of a right prism is the sum of the lateral area and the areas of the two bases. S.A. = L.A. + 2B Connection to Algebra The formula for the surface area of a cylinder is sometimes written as S.A. = 2pr(r + h). Have students show that this formula is equivalent to the formula S.A. = 2prh + 2pr2. 4 EXAMPLE 2 1 Finding Surface Area of a Cylinder Like a prism, a cylinder has two congruent parallel bases. However, the bases of a cylinder are circles. An altitude of a cylinder is a perpendicular segment that joins the planes of the bases. The height h of a cylinder is the length of an altitude. Diversity Some students may never have seen a steamroller. Have other students explain how steamrollers work. h Bases PowerPoint h h right cylinders Additional Examples oblique cylinder In this book you may assume that a cylinder is a right cylinder unless stated or pictured otherwise. 3 The radius of the base of a cylinder is 6 ft, and its height is 9 ft. Find its surface area in terms of p. 180π ft2 To ﬁnd the area of the curved surface of a cylinder, visualize “unrolling” it. The area of the resulting rectangle is the lateral area of the cylinder. The surface area of a cylinder is the sum of the lateral area and the areas of the two circular bases. You can ﬁnd formulas for these areas by looking at a net for a cylinder. r Real-World Connection A full turn of the roller inks a rectangle with area equal to the roller’s lateral area. Key Concepts Lateral Area h h Surface Area 2pr μ r r Area of a base B pr 2 Theorem 11-2 Lateral and Surface Areas of a Cylinder The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. B is the area of a base. r L.A. = 2prh, or L.A. = pdh The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. S.A. = L.A. + 2B, or S.A. = 2prh + 2pr2 610 610 Chapter 11 Surface Area and Volume h 3 Finding Surface Area of a Cylinder EXAMPLE The radius of the base of a cylinder is 4 in. and its height is 6 in. Find the surface area of the cylinder in terms of p. S.A. = L.A. + 2B = 2prh + Use the formula for surface area of a cylinder. 2(pr 2) Substitute the formulas for lateral area and area of a circle. = 2p(4)(6) + 2p(4 2) Substitute 4 for r and 6 for h. = 48p + 32p Simplify. = 80p Resources • Daily Notetaking Guide 11-2 The surface area of the cylinder is 80p in.2. Quick Check L3 3 Find the surface area of a cylinder with height 10 cm and radius 10 cm in terms of p. 400π cm2 4 Real-World EXAMPLE Connection The area covered is the lateral area of a cylinder that has a diameter of 4.2 ft and a height of 3.5 ft. Use the formula for lateral area of a cylinder. = p(4.2)(3.5) Substitute. = Use a calculator. 46.181412 In one full turn, the large drum covers about 46 ft2. Quick Check 4 The small drum has diameter 3 ft. 33 ft2 a. To the nearest square foot, what area does the small drum cover in one turn? b. Critical Thinking What area does the small drum cover in one turn of the large drum? same as large drum (about 46 ft2) EXERCISES • Daily Notetaking Guide 11-2— L1 Adapted Instruction Closure Machinery The drums of the roller at the left are cylinders of length 3.5 ft. The diameter of the large drum is 4.2 ft. What area does the large drum cover in one full turn? Round your answer to the nearest square foot. L.A. = pdh 4 A company sells cornmeal and barley in cylindrical containers. The diameter of the base of the 6-in. high cornmeal container is 4 in. The diameter of the base of the 4-in. high barley container is 6 in. Which container has the greater surface area? barley container Explain why the factor 2 appears in both terms of the formula for the surface area of a cylinder S.A. = 2prh + 2pr2. 2πrh is the lateral area, which is the area of a rectangle with one side equal to 2πr, the circumference of the base of the cylinder. The term 2πr2 is the sum of the areas of the two circular bases of the cylinder. For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A Practice by Example Example 1 GO for Help Use a net to ﬁnd the surface area of each prism. 1–3. See margin for drawings. 1. 2. 3. (page 609) 6 ft 29 cm 6 ft 4 in. 6 ft 216 ft2 6.5 cm 19 cm 1726 cm2 8 in. 4 in. 80 ± 32 "2 in.2 or about 125.3 in.2 4. a. Classify the prism. right hexagonal prism b. Find the lateral area of the prism. 240 cm2 c. The bases are regular hexagons. Find 10 cm the sum of their areas. 48 "3 cm2 4 cm d. Find the surface area of the prism. (240 ± 48 "3) cm2 Lesson 11-2 Surface Areas of Prisms and Cylinders 6.5 cm 1. 2. 3. 6 ft 6 ft 4 in. 4 in. 29 cm 6.5 cm 19 cm 6 ft 611 8 in. 4 "2 in. 4 "2 in. 611 3. Practice Example 2 (page 609) Use formulas to ﬁnd the lateral area and surface area of each prism. Show your answer to the nearest whole number. 880 cm2; 1121 cm2 5. Assignment Guide 8-15, 18, 22, 25, 27-32 C Challenge 33-37 Example 3 (page 611) 8. Example 4 (page 611) Exercise 22 Point out that the B coordinates are listed in the order (x, y, z). L4 L2 Reteaching L1 Adapted Practice Class L3 Date Practice 11-2 Chords and Arcs Apply Your Skills 18. A cylinder and a prism both have two O n bases and lateral faces that are rectangular. The bases of a cylinder are circles and the bases of a prism are polygons. 21d. 438 units2; 1752 units2; 4 : 1 0 Find the radius and m AB . 2. A 3. 2 cm C B A 5 cm C 6 in. B GO B 6 in. Find the value of x to the nearest tenth. 4. 5. 6. x C C x 5 2 C 7 3 8. 9. C x nline Homework Help Visit: PHSchool.com Web Code: aue-1102 x x 3 7. 12 7 3 8 x 14. 6 12 in. 612 © Pearson Education, Inc. All rights reserved. Q S R A U J C 21. Consider a box with dimensions 3, 4, and 5. a. Find its surface area. 94 units2 b. Double each dimension and then ﬁnd the new surface area. 376 units2 c. Find the ratio of the new surface area to the original surface area. 4 : 1 d. Repeat parts (a)–(c) for a box with dimensions 6, 9, and 11. See left. e. Make a Conjecture How does doubling the dimensions of a rectangular prism affect the surface area? The surface area becomes 4 times as large. Chapter 11 Surface Area and Volume 20. Answers may vary. Sample: L A 12. Prove Theorem 11-5, part (2). Given: 䉺O, OE # AB, OF # CD, AB = CD Prove: OE = OF 4 cm 3 cm E O B C F 0 0 0 13. Given: 䉺O with mAB = mBC = mCA Prove: m⬔ABC = m⬔BCA = m⬔CAB D A 14 cm O B C 612 1407 cm2 19. A hexagonal pencil is a hexagonal prism. A base edge of the pencil has length 4 mm. The pencil (without eraser) has height 170 mm. How much surface area of a hexagonal pencil gets painted? 4080 mm2 20. Open-Ended Draw a net for a rectangular prism with a surface area of 220 cm 2. See margin. 䉺A 䉺J, BC KL B K Write a two-column proof, a paragraph proof, or a ﬂow proof. 20 cm 17. Estimation Estimate the surface area of a cube with edges 4.95 cm long. 150 cm2 18. Writing Explain how a cylinder and a prism are alike and how they are different. See left. T P 8 cm 16. A triangular prism has base edges 4 cm, 5 cm, and 6 cm long. Its lateral area is 300 cm2. What is the height of the prism? 20 cm 20 11. 䉺Q 䉺T, PR SU 9m 226 m2 C 86ⴗ List what you can conclude from each diagram. 10. 15. 6m 6 C x 120ⴗ 11 in. 101.5π in.2 16.5π cm2 107 in.2 Exercise 26 Point out that the C 4 cm 4 in. Orange Juice Connection to Coordinate Geometry Enrichment 7 in. 12. Packaging A cylindrical carton of oatmeal with radius 3.5 in. is 9 in. tall. If all surfaces except the top are made of cardboard, how much cardboard is used to make the oatmeal carton? Round your answer to the nearest square inch. 236.4 in.2 Find the surface area of each cylinder to the nearest whole number. 13. surface area available for the label is in fact the lateral area of the can. L3 Regular octagon 11. A standard drinking straw is 19.5 cm long and has a diameter of 0.6 cm. How many square centimeters of plastic are used in one straw? Round your answer to the nearest tenth. 36.8 cm2 Exercise 2 After students finish the exercise, ask: What formula gives the surface area of a cube with sides of length s? S.A. ≠ 6s2 24 ft 5 ft 10. 3 cm 40π cm2 Connection to Algebra 1. A 9. 2 cm 8 cm Homework Quick Check GPS Guided Problem Solving 5 cm Find the surface area of each cylinder in terms of π. 38-41 42-46 To check students’ understanding of key skills and concepts, go over Exercises 4, 12, 18, 24, 26. 22 cm 5 in. 96 in.2; 108 in.2 10 ft 120 ft 2; 220 ft 2 2 A B Name 8 in. 5 ft 24, 26 Practice 3 in. 7. 4 in. 1 A B 1-7, 16, 17, 19-21, 23, Test Prep Mixed Review 6. 4 ft 4 cm 22. Multiple Choice A cylindrical can of cocoa has the dimensions shown at the right. What is the approximate surface area available for the label? A 110 in.2 148 in.2 2 179 in. 219 in.2 23. Pest Control A ﬂour moth trap has the shape of a triangular prism that is open on both ends. An environmentally safe chemical draws the moth inside the prism, which is lined with an adhesive. Find the surface area of the trap. 47.5 in.2 1 in. 7 in. C O C O A 5 in. 4. Assess & Reteach PowerPoint Lesson Quiz 2 in. 4 in. Use the prism below for Exercises 1 and 2. 5 in. 3.5 in. 24. Packaging A typical box for a videocassette tape is open on one side as GPS pictured at the left. How many square inches of cardboard are in a typical box for a videocassette tape? about 75.5 in.2 712 in. 4 in. Exercise 24 27. cylinder of radius 4 and height 2; 48π units2 25. Suppose that a cylinder has a radius of r units, and that the height of the cylinder is also r units. The lateral area of the cylinder is 98p square units. x 2 a. Algebra Find the value of r. 7 units b. Find the surface area of the cylinder. 196π units2 z 26. a. Geometry in 3 Dimensions Find the three D coordinates of each vertex A, B, C, and D 3 of the rectangular prism. A(3, 0, 0); B(3, 5, 0); 1 C(0, 5, 0); D(0, 5, 4) b. Find AB. 5 C 1 c. Find BC. 3 y O 2 4 d. Find CD. 4 3 B e. Find the surface area of the prism. 94 units2 x A 28. cylinder of radius 2 and height 4; 24π units2 Visualization The plane region is revolved completely about the given line to sweep out a solid of revolution. Describe the solid and ﬁnd its surface area in terms of π. 27–30. See left. 29. cylinder of radius 2 and height 4; 24π units2 27. the y-axis 28. the x-axis 29. the line y = 2 30. the line x = 4 30. cylinder of radius 4 and height 2; 48π units2 GO for Help For a guide to solving Exercise 32, see p. 615. 31b. Surface area is more than doubled. C Challenge 31c. S.A. ≠ ± 2πrh; if r doubles: S.A. ≠ 2(4πr 2 ± 2πrh). Since r is squared, surface area is more than doubled. 2πr 2 3 2 1 O m 7c 34. 4m x 1 2 3 4 8 cm (148 ± 66.5π) cm2 lesson quiz, PHSchool.com, Web Code: aua-1102 6m 3m (84 ± 20π) m2 1. Use a net to find the surface area. 8 ft 10 ft 8 ft 7 ft 6 ft 7 ft 6 ft 8 ft 7.5 in. 35. 8 in. 4. The radius of the base of a cylinder is 16 in., and its height is 4 in. Find its surface area in terms of p. 640π in.2 Alternative Assessment Have partners design and label two different prisms, each having a surface area of 200 cm2. 10 in. (220 – 8π) in.2 Lesson 11-2 Surface Areas of Prisms and Cylinders 3. The height of a prism is 5 cm. Its rectangular bases have 3-cm and 9-cm sides. Find its surface area. 174 cm2 5. A contractor paints all but the bases of a 28-ft high cylindrical water tank. The diameter of the base is 22 ft. How many square feet are painted? Round to the nearest hundred. 1900 ft2 6 in. 3 in. 4 cm 7 ft 2. Use a formula to find the surface area. S.A. ≠ L.A. ± 2B ≠ 168 ± 48 ≠ 216; 216 ft2 Judging by appearances, what is the surface area of each solid? 33. 6 ft S.A. ≠ 216 ft2 y 31. a. Critical Thinking Suppose you double the radius of a right cylinder. How does that affect the lateral area? Lateral area is doubled. b. How does that affect the surface area? b-c. See left below. c. Use the formula for surface area of a right cylinder to explain why the surface area in part (b) was not doubled. 32. a. Packaging Some cylinders have wrappers with a spiral seam. Peeled off, the wrapper has the shape of a parallelogram. The wrapper for a biscuit container has base 7.5 in. and height 6 in. Find the radius and height of the container. r < 1.2 in.; h ≠ 6 in. b. Find the surface area of the container. about 54 in.2 8 ft 613 613 Test Prep x 2 36. Algebra The sum of the height and radius of a cylinder is 9 m. The surface area of the cylinder is 54p m 2. Find the height and the radius. h ≠ 6 m; r ≠ 3 m Resources 37. Each edge of the large cube at the right is 12 inches long. The cube is painted on the outside, and then cut into 27 smaller cubes. Answer these questions about the 27 cubes. a. How many are painted on 4, 3, 2, 1, and 0 faces? b. What is the total surface area that is unpainted? b. 1728 in.2 a. 0, 8, 12, 6, 1 For additional practice with a variety of test item formats: • Standardized Test Prep, p. 657 • Test-Taking Strategies, p. 652 • Test-Taking Strategies with Transparencies Test Prep Multiple Choice 41. [2] a. Lateral area is the perimeter times height. Since the lateral area is 48 in.2 and the perimeter of the k is 24 in., h ≠ 2 in. b. S.A. ≠ 96 in.2 [1] correct explanation correct surface area Short Response 38. What is the surface area of the ﬁgure to the nearest tenth? C A. 335.7 m2 B. 411.6 m2 2 C. 671.5 m D. 721.2 m2 7.4 m 6.8 m 20.1 m 39. If the radius and height of a cylinder are both doubled, then the surface area is 9. J F. the same G. doubled H. tripled J. quadrupled 40. A cylinder of radius r sits snugly inside a cube. Which expression represents the difference of their lateral areas? D A. 2r2(8 - p) B. 2r(p - 2) C. 2r(4 - p) D. 4r2(4 - p) 41. The sides of a base of a right triangular prism are 6 in., 8 in., and 10 in. The lateral area of the prism is 48 in.2. a–b. See left. a. Find the height of the prism. Explain your reasoning. b. What is the surface area of the prism? Mixed Review Lesson 11-1 GO for Help Sketch each space ﬁgure and then draw a net for it. Label the net with its dimensions. 42. a rectangular box with height 5 cm and a base 3 cm by 4 cm 42–43. See back of book. 43. a cube with 2-in. sides Lesson 10-7 Find the area of each part of the circle to the nearest tenth. 44. sector QOP 37.7 cm2 Lesson 6-7 0 45. the segment of the circle bounded by QP and QP 22.1 cm2 46. In the kite at the right AB = AD and CB = CD. Points P, Q, R, and S are midpoints. a. Determine the coordinates of the R(c, b), S(c, –b), C(2c, 0) R midpoints. b. RQ = ■; SP = ■; P(a, –b), Q(a, b) PQ = ■; SR = ■ a – c; a – c; 2b; 2b c. Use your answers to part (b) to explain why PQRS must be a parallelogram. Opposite sides are O. 614 614 Chapter 11 Surface Area and Volume S P Q 120 cm O 6 B (0, 2b) Q O P D (0, -2b) A (2a, 0)