19.2 Surface Area of Prisms and Cylinders

Transcription

1 Name Class Date 19 Surface Area of Prisms and Cylinders Essential Question: How can you find te surface area of a prism or cylinder? Resource Locker Explore Developing a Surface Area Formula Surface area is te total area of all te faces and curved surfaces of a tree-dimensional figure Te lateral area of a prism is te sum of te areas of te lateral faces A Consider te rigt prism sown ere and te net for te rigt prism Complete te figure by labeling te dimensions of te net a c b b a c B C D E In te net, wat type of figure is formed by te lateral faces of te prism? rectangle Write an expression for te lengt of te base of te rectangle a + b + c How is te base of te rectangle related to te perimeter of te base of te prism? Tey are equal Te lateral area L of te prism is te area of te rectangle Write a formula for L in terms of, a, b, and c L = (a + b + c) F Write te formula for L in terms of P, were P is te perimeter of te base of te prism L = P Module Lesson DO NOT EDIT--Canges must be made troug File info CorrectionKey=NL-A;CA-A Date

2 G Let B be te area of te base of te prism Write a formula for te surface area S of te prism in terms of B and L Ten write te formula in terms of B, P, and S = L + B; S = P + B Reflect 1 Explain wy te net of te lateral surface of any rigt prism will always be a rectangle Sample answer: Eac lateral face of any rigt prism is a rectangle Te net of te lateral surface of any rigt prism is composed of rectangles joined end-to-end Straigt angles are formed wen te rectangles are joined in tis manner resulting in one long rectangular sape Suppose a rectangular prism as lengt l, widt w, and eigt, as sown Explain ow you can write a formula for te surface area of te prism in terms of l, w, and l w Sample answer: Tere are two faces wit area lw, two faces wit area w, and two faces wit area l, so te surface area can be written as S = lw + w + l Explain 1 Finding te Surface Area of a Prism Lateral Area and Surface Area of Rigt Prisms Te lateral area of a rigt prism wit eigt and base perimeter P is L = P Te surface area of a rigt prism wit lateral area L and base area B is S = L + B, or S = P + B B Module Lesson

3 Example 1 Eac gift box is a rigt prism Find te total amount of paper needed to wrap eac box, not counting overlap Step 1 Find te lateral area L = P P = (8) + (6) = 8 cm = 8(1) Multiply = 336 c m 1 cm Step Find te surface area Surface area formula S = L + B 8 cm 6 cm Substitute te lateral area = (6)(8) Simplify = 43 c m Step 1 Find te lengt c of te ypotenuse of te base Pytagorean Teorem c = a + b Substitute = in 4 in 0 in Image Credits: C Squared Studios/Potodisc/Getty Images Simplify = Take te square root of eac side c = 6 Step Find te lateral area L = P 60 0 Substitute = ( ) Multiply = 100 in Step 3 Find te surface area Surface area formula 676 S = L + B Substitute = _ Simplify = 1440 in 4 10 Reflect 3 A gift box is a rectangular prism wit lengt 98 cm, widt 10 cm, and eigt 97 cm Explain ow to estimate te amount of paper needed to wrap te box, not counting overlap Sample answer: Round eac dimension to 10 cm Ten eac face as an area of approximately 10 = 100 c m, and te surface area is approximately 6 (100) = 600 c m Module Lesson

4 Your Turn Eac gift box is a rigt prism Find te total amount of paper needed to wrap eac box, not counting overlap in 6 in 18 in 5 in 36 in 85 in Te lateral area is L = P P = (18) + (5) = 46 in So, L = 46 (5) = 30 i n Te surface area is S = L + B B = 18 (5) = 90 i n So, S = 30 + (90) = 410 i n Let b be te unknown lengt of te leg of te base By te Pytagorean Teorem, c = a + b, so 6 = b, 36 = b, and b = 304 Taking te square root of eac side sows tat b = 48 in Te lateral area is L = P P = = 144 in So, L = 144 (85) = 14 i n Te surface 1 area is S = L + B B = (48)(36) = 864 So, S = 14 + (864) = i n Explain Finding te Surface Area of a Cylinder Lateral Area and Surface Area of Rigt Cylinders Te lateral area of a cylinder is te area of te curved surface tat connects te two bases Te lateral area of a rigt cylinder wit radius r and eigt is L = πr Te surface area of a rigt cylinder wit lateral area L and base area B is S = L + B, or S = πr + π r r r πr Module Lesson

5 Example Eac aluminum can is a rigt cylinder Find te amount of paper needed for te can s label and te total amount of aluminum needed to make te can Round to te nearest tent Step 1 Find te lateral area L = πr 3 cm Substitute L = π (3) (9) Multiply = 54π cm 9 cm Step Find te surface area Surface area formula S = L + π r Substitute te lateral area and radius = 54π + π (3) Simplify = 7π cm Step 3 Use a calculator and round to te nearest tent Te amount of paper needed for te label is te lateral area, 54π 1696 c m Te amount of aluminum needed for te can is te surface area, 7π 6 c m Step 1 Find te lateral area L = πr Substitute; te radius is alf te diameter = π ( 5 )( ) Multiply = 10 π in Step Find te surface area in 5 in Reflect Surface area formula S = L + π r Substitute te lateral area and radius = 10 π + π ( 5 ) Simplify = 5 π in Step 3 Use a calculator and round to te nearest tent Te amount of paper needed for te label is te lateral area, 10 π 314 in Te amount of aluminum needed for te can is te surface area, 5 π 707 in 6 In tese problems, wy is it best to round only in te final step of te solution? Sample answer: Tis results in a more accurate answer If you round at an intermediate step, te inaccuracies may be compounded as you perform subsequent operations Module Lesson

6 Your Turn Eac aluminum can is a rigt cylinder Find te amount of paper needed for te can s label and te total amount of aluminum needed to make te can Round to te nearest tent mm Te lateral area is L = πr So, L = π (6)(15) = 180π cm Te surface area is S = L + πr So, S = 180π + π (6) = 5π cm 15 cm 6 cm Te amount of paper needed for te label is te lateral area, 180π 5655 c m Te amount of aluminum needed for te can is te surface area, 5π 7917 c m 7 mm Te radius of te cylinder is alf te diameter, so r = 36 mm Te lateral area is L = πr So, L = π (36)(80) = 5760π mm Te surface area is S = L + πr So, S = 5760π + π (36) = 835π mm Te amount of paper needed for te label is te lateral area, 5760π 18,0956 m m Te amount of aluminum needed for te can is te surface area, 835π 6,386 m m Explain 3 Finding te Surface Area of a Composite Figure Example 3 Find te surface area of eac composite figure Round to te nearest tent Step 1 Find te surface area of te rigt rectangular prism Surface area formula S = P + B 4 ft Substitute = 80 (0) + (4) (16) Simplify = 368 f t Step A cylinder is removed from te prism Find te lateral area of te cylinder and te area of its bases L = πr Substitute = π (4) (0) Simplify = 160π ft Base area formula B = π r Substitute = π (4) Simplify = 16π ft Step 3 Find te surface area of te composite figure Te surface area is te sum of te areas of all surfaces on te exterior of te figure S = (prism surface area) + (cylinder lateral area) - (cylinder base areas) = π - (16π) = π 7701 ft 4 ft 16 ft 0 ft Module Lesson

7 Step 1 Find te surface area of te rigt rectangular prism Surface area formula S = P + B Substitute = ( ) + ( ) ( ) Simplify = 0 cm Step Find te surface area of te cylinder 3 cm 9 cm cm 5 cm 4 cm L = πr Substitute = π ( )( 3 ) Simplify = 1 π cm Surface area formula S = L + π r Substitute = 1 π + π ( ) Simplify = 0 π cm Step 3 Find te surface area of te composite figure Te surface area is te sum of te areas of all surfaces on te exterior of te figure S = (prism surface area) + (cylinder surface area) - (area of one cylinder base) = π - π ( ) = π 397 cm Reflect 9 Discussion A student said te answer in Part A must be incorrect since a part of te rectangular prism is removed, yet te surface area of te composite figure is greater tan te surface area of te rectangular prism Do you agree wit te student? Explain No; removing part of te rectangular prism produces a ole troug te prism and tis creates additional exposed area on te interior surface of te ole Module Lesson

8 Your Turn Find te surface area of eac composite figure Round to te nearest tent in 3 in 7 mm 3 in 5 in 6 mm 9 in 7 in 3 mm Te surface area of te large prism is S large = P + B So, S large = (3) (5) + (9) (7) = 86 i n Te surface area of te small prism is S small = P + B So, S small = (16) (3) + (5) (3) = 78 i n Te surface area of te composite figure is te surface area of te large prism plus te surface area of te small prism minus times te area of te base of te small prism S = (5)(3) = 344 i n Te surface area of te large cylinder is S large = πr + πr So, S large = π (7) (6) + π (7) = 18π mm Te lateral area of te small prism is L small = πr So, L small = π (3) (6) = 36π mm Te area of eac base of te small cylinder is B = πr = π 3 = 9π mm Te surface area of te composite figure is te surface area of te large cylinder plus te lateral area of te small cylinder minus times te area of te base of te small cylinder S = 18π + 36π - (9π) = 00π 683 m m Elaborate 1 Can te surface area of a cylinder ever be less tan te lateral area of te cylinder? Explain No Te surface area is te lateral area plus te area of te two bases Since te area of te two bases is greater tan 0, te surface area must be greater tan te lateral area 13 Is it possible to find te surface area of a cylinder if you know te eigt and te circumference of te base? Explain Yes You can use te circumference of te base to find te radius of te base Ten you can use te eigt, circumference, and radius in te surface area formula 14 Essential Question Ceck-In How is finding te surface area of a rigt prism similar to finding te surface area of a rigt cylinder? In bot cases, you can find te surface area by finding te lateral area and ten adding twice te area of a base Module Lesson