Surface Areas of Prisms and Cylinders


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1 12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of a polyedron is te sum of te areas of its faces. Te lateral area of a polyedron is te sum of te areas of its lateral faces. Finding a Formula for Surface Area APPLYING MATHEMATICS To be proficient in mat, you need to analyze relationsips matematically to draw conclusions. Work wit a partner. Consider te polyedron sown. a. Identify te polyedron. Ten sketc its net so tat te lateral faces form a rectangle wit te same eigt as te polyedron. Wat types of figures make up te net? b. Write an expression tat represents te perimeter P of te base of te polyedron. Sow ow you can use P to write an expression tat represents te lateral area L of te polyedron. eigt, a b c c. Let B represent te area of a base of te polyedron. Write a formula for te surface area S. Finding a Formula for Surface Area Work wit a partner. Consider te solid sown. a. Identify te solid. Ten sketc its net. Wat types of figures make up te net? b. Write an expression tat represents te perimeter P of te base of te solid. Sow ow you can use P to write an expression tat represents te lateral area L of te solid. radius, r eigt, c. Write an expression tat represents te area B of a base of te solid. d. Write a formula for te surface area S. Communicate Your Answer 3. How can you find te surface area of a prism or a cylinder? 4. Consider te rectangular prism sown. a. Find te surface area of te rectangular prism by drawing its net and finding te sum of te areas of its faces. b. Find te surface area of te rectangular prism by using te formula you wrote in Exploration 1. c. Compare your answers to parts (a) and (b). Wat do you notice? Section 12.2 Surface Areas of Prisms and Cylinders 645
2 12.2 Lesson Wat You Will Learn Core Vocabulary lateral faces, p. 646 lateral edges, p. 646 surface area, p. 646 lateral area, p. 646 net, p. 646 rigt prism, p. 646 oblique prism, p. 646 rigt cylinder, p. 647 oblique cylinder, p. 647 Previous prism bases of a prism cylinder composite solid Find lateral areas and surface areas of rigt prisms. Find lateral areas and surface areas of rigt cylinders. Use surface areas of rigt prisms and rigt cylinders. Finding Lateral Areas and Surface Areas of Rigt Prisms Recall tat a prism is a polyedron wit two congruent faces, called bases, tat lie in parallel planes. Te oter faces, called lateral faces, are parallelograms formed by connecting te corresponding vertices of te bases. Te segments connecting tese vertices are lateral edges. Prisms are classified by te sapes of teir bases. base base lateral edges lateral faces Te surface area of a polyedron is te sum of te areas of its faces. Te lateral area of a polyedron is te sum of te areas of its lateral faces. Imagine tat you cut some edges of a polyedron and unfold it. Te twodimensional representation of te faces is called a net. Te surface area of a prism is equal to te area of its net. Te eigt of a prism is te perpendicular distance between its bases. In a rigt prism, eac lateral edge is perpendicular to bot bases. A prism wit lateral edges tat are not perpendicular to te bases is an oblique prism. eigt eigt Rigt rectangular prism Oblique triangular prism Core Concept Lateral Area and Surface Area of a Rigt Prism For a rigt prism wit base perimeter P, base apotem a, eigt, and base area B, te lateral area L and surface area S are as follows. L = P Surface area S = 2B + L = ap + P B P 646 Capter 12 Surface Area and Volume
3 Finding Lateral Area and Surface Area Find te lateral area and te surface area of te rigt pentagonal prism ft Find te apotem and perimeter of a base. 9 ft a = = P = 5(7.05) = a ATTENDING TO PRECISION Trougout tis capter, round lateral areas, surface areas, and volumes to te nearest undredt, if necessary. Find te lateral area and te surface area. L = P ft ft = (35.25)(9) Substitute. = Multiply. S = ap + P = ( ) (35.25) Substitute Formula for lateral area of a rigt prism Formula for surface area of a rigt prism Use a calculator. Te lateral area is square feet and te surface area is about square feet. Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com 1. Find te lateral area and te surface area of a rigt rectangular prism wit a eigt of 7 inces, a lengt of 3 inces, and a widt of 4 inces. eigt rigt cylinder eigt oblique cylinder Finding Lateral Areas and Surface Areas of Rigt Cylinders Recall tat a cylinder is a solid wit congruent circular bases tat lie in parallel planes. Te eigt of a cylinder is te perpendicular distance between its bases. Te radius of a base is te radius of te cylinder. In a rigt cylinder, te segment joining te centers of te bases is perpendicular to te bases. In an oblique cylinder, tis segment is not perpendicular to te bases. Te lateral area of a cylinder is te area of its curved surface. For a rigt cylinder, it is equal to te product of te circumference and te eigt, or 2πr. Te surface area of a cylinder is equal to te sum of te lateral area and te areas of te two bases. Core Concept Lateral Area and Surface Area of a Rigt Cylinder For a rigt cylinder wit radius r, r 2 πr 2 πr eigt, and base area B, te lateral area L and surface area S are as follows. Surface area L = 2πr S = 2B + L = 2πr 2 + 2πr r lateral area A = 2 r π base area A = πr 2 base area A = r 2 π Section 12.2 Surface Areas of Prisms and Cylinders 647
4 Finding Lateral Area and Surface Area Find te lateral area and te surface area of te rigt cylinder. 4 m Find te lateral area and te surface area. L = 2πr Formula for lateral area of a rigt cylinder = 2π(4)(8) Substitute. = 64π Simplify Use a calculator. S = 2πr 2 + 2πr Formula for surface area of a rigt cylinder = 2π(4) π Substitute. = 96π Simplify Use a calculator. 8 m Te lateral area is 64π, or about square meters. Te surface area is 96π, or about square meters. Solving a RealLife Problem You are designing a label for te cylindrical soup can sown. Te label will cover te lateral area of te can. Find te minimum amount of material needed for te label. 9 cm Find te radius of a base. r = 1 (9) = Find te lateral area. L = 2πr = 2π(4.5)(12) Substitute. = 108π Simplify Formula for lateral area of a rigt cylinder Use a calculator. 12 cm You need a minimum of about square centimeters of material. Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com 2. Find te lateral area and te surface area of te rigt cylinder. 10 in. 18 in. 3. WHAT IF? In Example 3, you cange te design of te can so tat te diameter is 12 centimeters. Find te minimum amount of material needed for te label. 648 Capter 12 Surface Area and Volume
5 Using Surface Areas of Rigt Prisms and Rigt Cylinders Finding te Surface Area of a Composite Solid 3 m 4 m Find te lateral area and te surface area of te composite solid. 12 m of solid = of cylinder + of prism = 2πr + P = 2π(6)(12) + 14(12) 6 m = 144π Surface area of solid = of solid + 2 ( Area of a base of te cylinder Area of a base of te prism ) = 144π (πr 2 w) = 144π [π(6) 2 4(3)] = 216π Te lateral area is about square meters and te surface area is about square meters. Canging Dimensions in a Solid Describe ow doubling all te linear dimensions affects te surface area of te rigt cylinder. 2 ft Before cange After cange Dimensions r = 2 ft, = 8 ft r = 4 ft, = 1 8 ft Surface area S = 2πr 2 + 2πr = 2π(2) 2 + 2π(2)(8) = 40π ft 2 S = 2πr 2 + 2πr = 2π(4) 2 + 2π(4)(16) = 160π ft 2 2 mm Doubling all te linear dimensions results in a surface area tat is 160π 40π = 4 = 22 times te original surface area. 8 mm 10 mm 6 mm Monitoring Progress Help in Englis and Spanis at BigIdeasMat.com 4. Find te lateral area and te surface area of te composite solid at te left. 5. In Example 5, describe ow multiplying all te linear dimensions by 1 affects te 2 surface area of te rigt cylinder. Section 12.2 Surface Areas of Prisms and Cylinders 649
6 12.2 Exercises Dynamic Solutions available at BigIdeasMat.com Vocabulary and Core Concept Ceck 1. VOCABULARY Sketc a rigt triangular prism. Identify te bases, lateral faces, and lateral edges. 2. WRITING Explain ow te formula S = 2B + L applies to finding te surface area of bot a rigt prism and a rigt cylinder. Monitoring Progress and Modeling wit Matematics In Exercises 3 and 4, find te surface area of te solid formed by te net in. 13. MODELING WITH MATHEMATICS Te inside of te cylindrical swimming pool sown must be covered wit a vinyl liner. Te liner must cover te side and bottom of te swimming pool. Wat is te minimum amount of vinyl needed for te liner? (See Example 3.) 24 ft 10 in. 20 cm 4 ft In Exercises 5 8, find te lateral area and te surface area of te rigt prism. (See Example 1.) ft 8 ft 3 ft 3 m 8 m 9.1 m 7. A regular pentagonal prism as a eigt of 3.5 inces and a base edge lengt of 2 inces. 8. A regular exagonal prism as a eigt of 80 feet and a base edge lengt of 40 feet. In Exercises 9 12, find te lateral area and te surface area of te rigt cylinder. (See Example 2.) in in. 16 cm 14. MODELING WITH MATHEMATICS Te tent sown as fabric covering all four sides and te floor. Wat is 4 ft te minimum amount of fabric needed to construct te tent? In Exercises 15 18, find te lateral area and te surface area of te composite solid. (See Example 4.) cm 4 cm 1 cm cm 4 ft 5 ft 8 ft 7 ft 4 ft 11. A rigt cylinder as a diameter of 24 millimeters and a eigt of 40 millimeters in m 7 m 12. A rigt cylinder as a radius of 2.5 feet and a eigt of 7.5 feet. 11 in. 9 m 6 m 15 m 5 in. 650 Capter 12 Surface Area and Volume
7 19. ERROR ANALYSIS Describe and correct te error in finding te surface area of te rigt cylinder. 6 cm S = 2π (6) 2 + 2π(6)(8) = 168π cm ERROR ANALYSIS Describe and correct te error in finding te surface area of te composite solid. 1 7 ft 27. MATHEMATICAL CONNECTIONS A cube as a surface area of 343 square inces. Write and solve an equation to find te lengt of eac edge of te cube. 28. MATHEMATICAL CONNECTIONS A rigt cylinder as a surface area of 108π square meters. Te radius of te cylinder is twice its eigt. Write and solve an equation to find te eigt of te cylinder. 29. MODELING WITH MATHEMATICS A company makes two types of recycling bins, as sown. Bot types of bins ave an open top. Wic recycling bin requires more material to make? Explain. 6 in. 20 ft 18 ft S = 2(20)(7) + 2(18)(7) + 2π (8)(7) + 2[(18)(20) + π (8) 2 ] ft 2 36 in. 36 in. In Exercises 21 24, describe ow te cange affects te surface area of te rigt prism or rigt cylinder. (See Example 5.) 21. doubling all te linear dimensions 17 in. 5 in. 4 in. 22. multiplying all te linear dimensions by mm 10 in. 12 in. 30. MODELING WITH MATHEMATICS You are painting a rectangular room tat is 13 feet long, 9 feet wide, and 8.5 feet ig. Tere is a window tat is 2.5 feet wide and 5 feet ig on one wall. On anoter wall, tere is a door tat is 4 feet wide and 7 feet ig. A gallon of paint covers 350 square feet. How many gallons of paint do you need to cover te four walls wit one coat of paint, not including te window and door? 23. tripling te radius 2 yd 7 yd 24 mm 24. multiplying te base edge lengts by 1 4 and te eigt by 4 2 m 8 m 16 m 31. ANALYZING RELATIONSHIPS Wic creates a greater surface area, doubling te radius of a cylinder or doubling te eigt of a cylinder? Explain your reasoning. 32. MAKING AN ARGUMENT You cut a cylindrical piece of lead, forming two congruent cylindrical pieces of lead. Your friend claims te surface area of eac smaller piece is exactly alf te surface area of te original piece. Is your friend correct? Explain your reasoning. In Exercises 25 and 26, find te eigt of te rigt prism or rigt cylinder. 25. S = 1097 m S = 480 in m 8 in. 15 in. 33. USING STRUCTURE Te rigt triangular prisms sown ave te same surface area. Find te eigt of prism B. 20 cm Prism A 24 cm 20 cm 3 cm 6 cm Prism B Section 12.2 Surface Areas of Prisms and Cylinders 651
8 34. USING STRUCTURE Te lateral surface area of a regular pentagonal prism is 360 square feet. Te eigt of te prism is twice te lengt of one of te edges of te base. Find te surface area of te prism. 35. ANALYZING RELATIONSHIPS Describe ow multiplying all te linear dimensions of te rigt rectangular prism by eac given value affects te surface area of te prism. 38. THOUGHT PROVOKING You ave 24 cubesaped building blocks wit edge lengts of 1 unit. Wat arrangement of blocks gives you a rectangular prism wit te least surface area? Justify your answer. 39. USING STRUCTURE Sketc te net of te oblique rectangular prism sown. Ten find te surface area. 4 ft 8 ft 7 ft a. 2 b. 3 c. 1 2 d. n 36. HOW DO YOU SEE IT? An open gift box is sown. a. Wy is te area of te net of te box larger tan te minimum amount of wrapping paper needed dd to cover te closed box? b. Wen wrapping te box, wy would you want to use more tan te minimum amount of paper needed? 37. REASONING Consider a cube tat is built using 27 unit cubes, as sown. a. Find te surface area of te solid formed wen te red unit cubes are removed from te solid sown. b. Find te surface area of te solid formed wen te blue unit cubes are removed from te solid sown. c. Explain wy your answers are different in parts (a) and (b). w 15 ft 40. WRITING Use te diagram to write a formula tat can be used to find te surface area S of any cylindrical ring were 0 < r 2 < r 1. r USING STRUCTURE Te diagonal of a cube is a segment wose endpoints are vertices tat are not on te same face. Find te surface area of a cube wit a diagonal lengt of 8 units. 42. USING STRUCTURE A cuboctaedron as 6 square faces and 8 equilateral triangular faces, as sown. A cuboctaedron can be made by slicing off te corners of a cube. a. Sketc a net for te cuboctaedron. b. Eac edge of a cuboctaedron as a lengt of 5 millimeters. Find its surface area. r 2 Maintaining Matematical Proficiency Reviewing wat you learned in previous grades and lessons Find te area of te regular polygon. (Section 11.3) in m 9 in. 6 cm 6.2 cm 652 Capter 12 Surface Area and Volume
12 Surface Area and Volume
12 Surface Area and Volume 12.1 ThreeDimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids
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