11-2 11-2 Surface Areas of Prisms 11-2 and Cylinders 1. Plan
Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lessons 1-9 and 10-3 1 To find the surface area of a • To find the surface area of a prism Find the area of each net. 40π cm2 363 m2 prism " 2 To find the surface area of a 1.2 2. 3. • To find the surface area of a 96 cm 4 cm cylinder cylinder 4 cm Examples . . . And Why 8 cm 6 m 1 Finding Surface Area of 4 cm a Prism To find the area covered by a drum on a roller used in road 2 Using Formulas to Find construction, as in Example 4 Surface Area 4 cm 3 Finding Surface Area of a Cylinder New Vocabulary • prism • bases, lateral faces, altitude, height, lateral area, 4 Real-World Connection 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 1 Finding Surface Area of a Prism develop formulas for the lateral and surface areas of prisms and Aprism is a polyhedron with Lateral cylinders. Nets especially simplify exactly two congruent, parallel edges finding these areas for nonright faces, called bases. Other faces prisms, such as parallelepipeds. are lateral faces. You name a The key idea is that the lateral prism by the shape of its bases. Bases Bases faces of any prism are Lateral parallelograms. An altitude of a prism is a perpendicular segment that faces More Math Background: p. 596C joins the planes of the bases. The height h of the prism is Pentagonal prism Triangular prism the length of an altitude. Lesson Planning and A prism may either be right or oblique. Resources Real-World Connection h See p. 596E for a list of the h A triangular prism h resources that support this lesson. breaks white light into rainbow colors. right prisms oblique prism PowerPoint In a right prism the lateral faces are rectangles and a lateral edge is an altitude. Bell Ringer Practice In this book you may assume that a prism is a right prism unless stated or Check Skills You’ll Need pictured otherwise. For intervention, direct students to: Finding Area The lateral area of a prism is the sum of the areas of the lateral faces. The Lesson 1-9: Examples 4–6 surface area is the sum of the lateral area and the area of the two bases. Extra Skills, Word Problems, Proof Practice, Ch. 1
Areas of Regular Polygons 608 Chapter 11 Surface Area and Volume Lesson 10-3: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 10 Special Needs L1 Below Level L2 Draw examples of prisms on the board and have Making nets of rectangular prisms may help students students identify the bases, lateral faces, and understand and remember Theorem 11-1. altitudes. Point out that in a rectangular prism, any pair of opposite faces could be considered the bases. 608 learning style: visual learning style: tactile 1 EXAMPLE Finding Surface Area of a Prism 2. Teach Use a net to find the surface area of the prism at the left. 4 Surface Area = Lateral Area + area of bases 3 4433 Guided Instruction = sum of areas of lateral faces + area of bases 5 = (5 ? 4 + 5 ? 3 + 5 ? 4 + 5 ? 3) + 2(3)(4) 5 cm 3 Error Prevention! = 70 + 24 4 Some students may think that a 3 cm = 94 right prism must contain a right 4 cm The surface area of the prism is 94 cm2. angle in its base. Point out that the right angle is formed by the Quick Check 1 Use a net to find the surface area lateral faces meeting the base. 5 cm 5 cm of the triangular prism. See left. 12 cm 2 EXAMPLE Alternative Method 1. 216 cm2 6 cm Have students calculate the sum 5 cm You can find formulas for lateral and surface areas by looking at a net for a prism. of the areas of the lateral faces Perimeter of base c to help them understand why the c = a b c d ddBase formula L.A. ph makes sense. 12 cm a b c daa b bcd h PowerPoint hh 6 cm Additional Examples d Base Lateral Area of 1 Use a net to find the surface Perimeter Height c Area a base area of the cube. Lateral Area ph Surface Area L.A. 2B
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? 726 in.2 72 cm2 78 cm2 84 cm2 96 cm2 D E B C 1 A D E B C 2 A D E B C 2 D E Find the surface area of a 3 A B C 4 A D E By the Pythagorean Theorem, the hypotenuse of the B C 3 cm 4 cm 5 A D E B C 10-cm high right prism with Test-Taking Tip triangular base is 5 cm. triangular bases having 18-cm A question could ask L.A. = ph Use the formula for lateral area. edges. Round to the nearest for either surface area 2 = ? p ≠ 3 ± 4 ± 5 ≠ 12 cm whole number. 821 cm or lateral area of a 12 6 6 cm solid. Read the = 72 question carefully. The lateral area of the prism is 72 cm2. Now use the formula for surface area. S.A. = L.A. + 2B = + = ≠ 1 ? ≠ 2 72 2(6) 84 B 2 (3 4) 6 cm 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 6 m
Lesson 11-2 Surface Areas of Prisms and Cylinders 609
Advanced Learners L4 English Language Learners ELL After Example 2, ask students to write a formula for To help students with the new vocabulary, have them the surface area of a rectangular prism with edges of make and display a poster of a prism and a cylinder, length <, w, and h. S.A. ≠ 2(h< ± hw ±
PowerPoint right cylinders oblique cylinder Additional Examples 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 To find the area of the curved surface of a cylinder, visualize “unrolling” it. The 9 ft. Find its surface area in terms area of the resulting rectangle is the lateral area of the cylinder. The surface area of p. 180π ft2 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 r
Lateral Surface h h Real-World Connection Area Area μ A full turn of the roller inks a 2pr r rectangle with area equal to Area of a base the roller’s lateral area. B pr2
Key Concepts Theorem 11-2 Lateral and Surface Areas of a Cylinder
The lateral area of a right cylinder is the product of the B is the area circumference of the base and the height of the cylinder. of a base. r L.A. = 2prh, or L.A. = pdh h 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 Chapter 11 Surface Area and Volume
610 4 A company sells cornmeal and 3 EXAMPLE Finding Surface Area of a Cylinder barley in cylindrical containers. The diameter of the base of the The radius of the base of a cylinder is 4 in. and its height is 6 in. Find the surface 6-in. high cornmeal container is area of the cylinder in terms of p. 4 in. The diameter of the base of S.A. = L.A. + 2B Use the formula for surface area of a cylinder. the 4-in. high barley container = 2prh + 2(pr2) Substitute the formulas for lateral area and area of a circle. is 6 in. Which container has the greater surface area? barley = + 2 2p(4)(6) 2p(4 ) Substitute 4 for r and 6 for h. container = 48p + 32p Simplify. = 80p Resources The surface area of the cylinder is 80p in.2. • Daily Notetaking Guide 11-2 L3 Quick Check 3 Find the surface area of a cylinder with height 10 cm and radius 10 cm in terms of p. • Daily Notetaking Guide 11-2— L1 400π cm2 Adapted Instruction
4 EXAMPLE Real-World Connection 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 Explain why the factor 2 appears full turn? Round your answer to the nearest square foot. in both terms of the formula for The area covered is the lateral area of a cylinder that has a diameter of 4.2 ft and a the surface area of a cylinder = p + p 2 2πrh is the height of 3.5 ft. S.A. 2 rh 2 r . lateral area, which is the area of L.A. = pdh Use the formula for lateral area of a cylinder. a rectangle with one side equal = p(4.2)(3.5) Substitute. to 2πr, the circumference of the base of the cylinder. The term = 46.181412 Use a calculator. 2πr2 is the sum of the areas of In one full turn, the large drum covers about 46 ft2. the two circular bases of the cylinder. 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 For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving
A Practice by Example Use a net to find the surface area of each prism. 1–3. See margin for drawings. Example 1 1. 2. 3. for (page 609) 6 ft GO 29 cm Help 6 ft 4 in. 6 ft 8 in. 2 6.5 cm 216 ft 4 in. 19 cm 2 1726 cm 80 ± 32 2 in.2 or " 4. a. Classify the prism.right hexagonal prism about 125.3 in.2 b. Find the lateral area of the prism. 240 cm2 c. The bases are regular hexagons. Find the sum of their areas. 483 cm2 10 cm " 4 cm d. Find the surface area of the prism. (240 ± 483 ) cm2 " Lesson 11-2 Surface Areas of Prisms and Cylinders 611
1.6.5 cm 2. 3. 6 ft 4 in. 6 ft 4 in. 4 2 in. 29 cm " 4 2 in. " 6.5 cm 8 in. 6 ft 19 cm 611 3. Practice Example 2 Use formulas to find the lateral area and surface area of each prism. Show your (page 609) answer to the nearest whole number. 880 cm2; 1121 cm2 5. 6. 7. Assignment Guide 4 ft 3 in. 4 in. 1 AB1-7, 16, 17, 19-21, 23, 8 in. 5 ft 22 cm 24, 26 10 ft 5 in. 120 ft2; 220 ft2 96 in.2; 108 in.2 5 cm Regular 2 AB 8-15, 18, 22, 25, octagon 27-32 Example 3 Find the surface area of each cylinder in terms of π. C Challenge 33-37 (page 611) 8.2 cm 9.3 cm 10. 7 in. Test Prep 38-41 8 cm Mixed Review 42-46 4 cm Homework Quick Check 11 in. 2 To check students’ understanding 40π cm2 101.5π in. 16.5π cm2 of key skills and concepts, go over Exercises 4, 12, 18, 24, 26. 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 2 Connection to Algebra to the nearest tenth. 36.8 cm Exercise 2 After students finish Example 4 12. Packaging A cylindrical carton of oatmeal with radius 3.5 in. is 9 in. tall. If all the exercise, ask: What formula (page 611) surfaces except the top are made of cardboard, how much cardboard is used to gives the surface area of a cube make the oatmeal carton? Round your answer to the nearest square inch. with sides of length s? S.A. ≠ 6s2 236.4 in.2 Find the surface area of each cylinder to the nearest whole number. Exercise 22 Point out that the
surface area available for the 13.4 in. 14. 15. 8 cm
label is in fact the lateral area of e c
i 1 the can. e u 6 in. g 2 20 cm
J n 6 m 9 m
a r
Connection to O 226 m2 1407 cm2 Coordinate Geometry 107 in.2 Exercise 26 Point out that the B Apply Your Skills 16. A triangular prism has base edges 4 cm, 5 cm, and 6 cm long. Its lateral area coordinates are listed in the is 300 cm2. What is the height of the prism? 20 cm order (x, y, z). 17. Estimation Estimate the surface area of a cube with edges 4.95 cm long. 150 cm2 18. A cylinder and a 18. Writing Explain how a cylinder and a prism are alike and how they prism both have two are different. See left. Onbases and lateral faces that are 19. A hexagonal pencil is a hexagonal prism. A base rectangular. The edge of the pencil has length 4 mm. The pencil GPS Guided Problem Solving L3 bases of a cylinder (without eraser) has height 170 mm. How much Enrichment L4 are circles and the surface area of a hexagonal pencil gets painted? bases of a prism are 2 Reteaching L2 4080 mm polygons. 20. Open-Ended Draw a net for a rectangular Adapted Practice L1 prism with a surface area of 220 cm2. See margin. 21d. 438 units2; PracticeName Class Date L3 1752 units2; 4 : 1 Practice 11-2 Chords and Arcs 21. Consider a box with dimensions 3, 4, and 5. 0 2 Find the radius and mAB . a. Find its surface area. 94 units A 1.A 2. 3. 24 ft 5 ft C 2 cm 2 5 cm B C b. Double each dimension and then find the new surface area. 376 units C 6 in.
A B B 6 in. GO nline c. Find the ratio of the new surface area to the original surface area. 4:1 Find the value of x to the nearest tenth. 4. 5. 6. Homework Help x d. Repeat parts (a)–(c) for a box with dimensions 6, 9, and 11. See left. C 2 C C x 77 5 3 3 x e. Make a Conjecture How does doubling the dimensions of a rectangular 8 3 x Visit: PHSchool.com
7. 8. 9. x Web Code: aue-1102 prism affect the surface area? The surface area becomes 4 times as large. C x 12 C 6 x C 20
120؇ 86؇ 612 Chapter 11 Surface Area and Volume List what you can conclude from each diagram.
10.᭪Q ᭪T, PR SU 11. ᭪A ᭪J, BC KL B K Q T P S A J R U C L 20. Answers may vary. 4 cm Write a two-column proof, a paragraph proof, or a flow proof. 3 cm 12. Prove Theorem 11-5, part (2). A Sample: Given: ᭪O, OE # AB, OF # CD,AB= CD E = O Prove: OE OF B C © Pearson Education, Inc. All rights reserved. F D 0 0 0 13. Given: ᭪O with mAB = mBC = mCA A Prove: mЄABC = mЄBCA = mЄCAB 14 cm O B
C
612 4 cm 22. Multiple Choice A cylindrical can of cocoa has the 7 in. dimensions shown at the right. What is the 4. Assess & Reteach approximate surface area available for the label? A C 5 in. 110 in.2 148 in.2 O C O A PowerPoint 179 in.2 219 in.2 Lesson Quiz 23. Pest Control A flour moth trap has the shape of a triangular prism that is open on both ends. An Use the prism below for Exercises environmentally safe chemical draws the moth 4 in. 1 and 2. 2 in. 5 in. inside the prism, which is lined with an adhesive. 3.5 in. Find the surface area of the trap. 47.5 in.2 1 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 8 ft for a videocassette tape? about 75.5 in.2 25. Suppose that a cylinder has a radius of r units, and that the height of the 1 72 in. cylinder is also r units. The lateral area of the cylinder is 98p square units. 6 ft 7 ft x 2 a. Algebra Find the value of r. 7 units b. Find the surface area of the cylinder. 196π units2 1. Use a net to find the surface z area. 26. a. Geometry in 3 Dimensions Find the three D 4 in. coordinates of each vertex A, B, C, and D 3 8 ft Exercise 24 of the rectangular prism. A(3, 0, 0); B(3, 5, 0); 8 ft 10 ft b. Find AB. 5 C(0, 5, 0); D(0, 5, 4) 1 C 6 ft c. Find BC. 3 1 7 ft 7 ft O 24 y 6 ft d. Find CD. 4 3 27. cylinder of radius 4 2 A B e. Find the surface area of the prism. 94 units x 8 ft and height 2; 48π units2 Visualization The plane region is revolved ≠ 2 28. cylinder of radius 2 completely about the given line to sweep out y S.A. 216 ft and height 4; a solid of revolution. Describe the solid and 3 2. Use a formula to find 2 2 24π units find its surface area in terms of π. 27–30. See left. 1 the surface area. 29. cylinder of radius 2 S.A. ≠ L.A. ± 2B ≠ 27. the y-axis 28. the x-axis and height 4; O 1 234 x 168 ± 48 ≠ 216; 216 ft2 24π units2 = = 29. the line y 2 30. the line x 4 3. The height of a prism is 30. cylinder of radius 4 5 cm. Its rectangular bases Critical Thinking and height 2; 31. a. Suppose you double the radius of a right cylinder. How have 3-cm and 9-cm sides. Find 48π units2 Lateral area is doubled. does that affect the lateral area? its surface area. 174 cm2 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 4. The radius of the base surface area in part (b) was not doubled. of a cylinder is 16 in., and its height is 4 in. Find its surface GO for Help 32. a. Packaging Some cylinders have wrappers area in terms of p. 640π in.2 with a spiral seam. Peeled off, the wrapper 5. A contractor paints all but the For a guide to solving has the shape of a parallelogram. The 6 in. bases of a 28-ft high cylindrical Exercise 32, see p. 615. wrapper for a biscuit container has base water tank. The diameter of 7.5 in. and height 6 in. Find the radius and the base is 22 ft. How many 31b. Surface area is height of the container. r < 1.2 in.; h ≠ 6 in. more than doubled. 2 7.5 in. square feet are painted? b. Find the surface area of the container. about 54 in. Round to the nearest hundred. 1900 ft2 C Challenge Judging by appearances, what is the surface area of each solid? 31c. S.A. ≠ 2 r2 ± 2 rh; π π 33. 34.4 m 35. if r doubles: 8 in. Alternative Assessment S.A. ≠ 2(4πr2 ± 7 cm 2πrh). Since r is 3 in. Have partners design and label squared, surface 4 cm two different prisms, each having area is more than 8 cm 6 m 3 m 10 in. (84 ± 20π) m2 (220 – 8π) in.2 a surface area of 200 cm2. doubled. (148 ± 66.5π) cm2 lesson quiz, PHSchool.com, Web Code: aua-1102 Lesson 11-2 Surface Areas of Prisms and Cylinders 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 m2. Find the height and the radius. h ≠ 6 m; r ≠ 3 m Resources For additional practice with a 37. Each edge of the large cube at the right is variety of test item formats: 12 inches long. The cube is painted on the • Standardized Test Prep, p. 657 outside, and then cut into 27 smaller cubes. • Test-Taking Strategies, p. 652 Answer these questions about the 27 cubes. • Test-Taking Strategies with a. How many are painted on 4, 3, 2, 1, and 0 faces? Transparencies b. What is the total surface area that is unpainted? a. 0, 8, 12, 6, 1 b. 1728 in.2
Test Prep
Multiple Choice 38. What is the surface area of the figure 7.4 m 41. [2] a. Lateral area is to the nearest tenth? C 2 2 the perimeter A. 335.7 m B. 411.6 m 6.8 m times height. C. 671.5 m2 D. 721.2 m2 Since the lateral 20.1 m area is 48 in.2 and 39. If the radius and height of a cylinder are 9 the perimeter of both doubled, then the surface area is . J the k is 24 in., F. the same G. doubled H. tripled J. quadrupled h ≠ 2 in. 40. A cylinder of radius r sits snugly inside a cube. Which expression represents ≠ 2 b. S.A. 96 in. the difference of their lateral areas? D [1] correct explanation A. 2r2(8 - p) B. 2r(p - 2) C. 2r(4 - p) D. 4r2(4 - p) correct surface area Short Response 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?
MixedMixed ReviewReview
Lesson 11-1 Sketch each space figure and then draw a net for it. Label the net with for its dimensions. GO Help 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. Q P 120 2 44. sector QOP 37.7 cm 6 cm 0 O 45. the segment of the circle bounded by QP and QP 22.1 cm2 Lesson 6-7 46. In the kite at the right AB = AD and CB = CD. Points P, Q, R, and S are midpoints. B(0, 2b) a. Determine the coordinates of the R Q midpoints. R(c, b), S(c, –b), – C(2c, 0)O A (2a, 0) b. RQ = ■; SP = ■; P(a, b), Q(a, b) = ■ = ■ – – PQ ; SR a c; a c; 2b; 2b S P c. Use your answers to part (b) to explain D (0, -2b) why PQRS must be a parallelogram. Opposite sides are O.
614 Chapter 11 Surface Area and Volume
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