17.2 Surface Area of Prisms and Cylinders

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1 Name Class Date 17. Surface Area of Prisms and Cylinders Essential Question: How can you find the surface area of a prism or cylinder? Explore G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms,... cylinders,... to solve problems using appropriate units of measure. Also G.10.B Developing a Surface Area Formula Resource Locker Surface area is the total area of all the faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. Consider the right prism shown here and the net for the right prism. Complete the figure by labeling the dimensions of the net. h a c b B In the net, what type of figure is formed by the lateral faces of the prism? Write an expression for the length of the base of the rectangle. How is the base of the rectangle related to the perimeter of the base of the prism? The lateral area L of the prism is the area of the rectangle. Write a formula for L in terms of h, a, b, and c. Module Lesson

2 F Write the formula for L in terms of P, where P is the perimeter of the base of the prism. G Let B be the area of the base of the prism. Write a formula for the surface area S of the prism in terms of B and L. Then write the formula in terms of B, P, and h. Reflect 1. Explain why the net of the lateral surface of any right prism will always be a rectangle.. Suppose a rectangular prism has length l, width w, and height h, as shown. Explain how you can write a formula for the surface area of the prism in terms of l, w, and h. h l w Explain 1 Finding the Surface Area of a Prism Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with height h and base perimeter P is L = Ph. The surface area of a right prism with lateral area L and base area B is S = L + B, or S = Ph + B. h B Module Lesson

3 Example 1 Each gift box is a right prism. Find the total amount of paper needed to wrap each box, not counting overlap. Step 1 Find the lateral area. Lateral area formula L = Ph P = (8) + (6) = 8 cm = 8(1) Multiply. = 336 c m Step Find the surface area. Surface area formula S = L + B Substitute the lateral area. = (6)(8) Simplify. = 43 c m 8 cm 1 cm 6 cm B Image Credits: C Squared Studios/Photodisc/Getty Images Step 1 Find the length c of the hypotenuse of the base. Pythagorean Theorem c = a + b Substitute. = + Simplify. = Take the square root of each side. c = Step Find the lateral area. Lateral area formula L = Ph Substitute. = ( ) Multiply. = in 10 in. 4 in. 0 in. Module Lesson

4 Step 3 Find the surface area. Surface area formula S = L + B Substitute. = + 1_ Simplify. = i n Reflect 3. A gift box is a rectangular prism with length 9.8 cm, width 10. cm, and height 9.7 cm. Explain how to estimate the amount of paper needed to wrap the box, not counting overlap. Your Turn Each gift box is a right prism. Find the total amount of paper needed to wrap each box, not counting overlap in. 6 in. 18 in. 5 in. 3.6 in. 8.5 in. Module Lesson

5 Explain Finding the Surface Area of a Cylinder Lateral Area and Surface Area of Right Cylinders The lateral area of a cylinder is the area of the curved surface that connects the two bases. The lateral area of a right cylinder with radius r and height h is L = πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + B, or S = πrh + πr. r r πr h h Example Each aluminum can is a right cylinder. Find the amount of paper needed for the can s label and the total amount of aluminum needed to make the can. Round to the nearest tenth. Step 1 Find the lateral area. 3 cm Lateral area formula L = πrh Substitute. L = π (3) (9) Multiply. = 54π cm Step Find the surface area. Surface area formula S = L + π r Substitute the lateral area and radius. = 54π + r (3) Simplify. = 7π cm Step 3 Use a calculator and round to the nearest tenth. 9 cm The amount of paper needed for the label is the lateral area, 54π c m. The amount of aluminum needed for the can is the surface area, 7π 6. c m. Module Lesson

6 B Step 1 Find the lateral area. Lateral area formula L = πrh in 5 in Substitute; the radius is half the diameter. = π ( ) ( ) Multiply. = π in Step Find the surface area. Surface area formula S = L + π r Substitute the lateral area and radius. = π + r ( ) Simplify. = π in Step 3 Use a calculator and round to the nearest tenth. The amount of paper needed for the label is the lateral area, π i n. The amount of aluminum needed for the can is the surface area, π i n. Reflect 6. In these problems, why is it best to round only in the final step of the solution? Your Turn Each aluminum can is a right cylinder. Find the amount of paper needed for the can s label and the total amount of aluminum needed to make the can. Round to the nearest tenth mm 15 cm 6 cm 7 mm Module Lesson

7 Explain 3 Finding the Surface Area of a Composite Figure Example 3 Find the surface area of each composite figure. Round to the nearest tenth. Step 1 Find the surface area of the right rectangular prism. Surface area formula S = Ph + B 4 ft 0 ft Substitute. = 80 (0) + (4) (16) Simplify. = 368 f t Step A cylinder is removed from the prism. Find the lateral area of the cylinder and the area of its bases. 4 ft 16 ft Lateral area formula L = πrh Substitute. = π (4) (0) Simplify. = 160π ft Base area formula B = π r Substitute. = π (4) Simplify. = 16π ft Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder lateral area) - (cylinder base areas) = π - (16π) = π ft Module Lesson

8 B Step 1 Find the surface area of the right rectangular prism. 3 cm cm Surface area formula S = Ph + B Substitute. = ( ) + ( ) ( ) Simplify. = cm Step Find the surface area of the cylinder. Lateral area formula L = πrh Substitute. = π ( ) ( ) Simplify. = π cm Surface area formula S = L + π r Substitute. = π + π ( ) Simplify. = π cm 9 cm 5 cm 4 cm Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder surface area) - (area of one cylinder base) = + π - π ( ) = + π cm Reflect 9. Discussion A student said the answer in Part A must be incorrect since a part of the rectangular prism is removed, yet the surface area of the composite figure is greater than the surface area of the rectangular prism. Do you agree with the student? Explain. Module Lesson

9 Your Turn Find the surface area of each composite figure. Round to the nearest tenth in 3 in 7 mm 3 in 5 in 6 mm 9 in 7 in 3 mm Elaborate 1. Can the surface area of a cylinder ever be less than the lateral area of the cylinder? Explain. 13. Is it possible to find the surface area of a cylinder if you know the height and the circumference of the base? Explain. 14. Essential Question Check-In How is finding the surface area of a right prism similar to finding the surface area of a right cylinder? Module Lesson

10 Evaluate: Homework and Practice Find the lateral area and surface area of each prism cm 3 cm Online Homework Hints and Help Extra Practice 3 ft 7 ft 5 cm cm 5 ft cm 10 cm 5 cm 5 cm 1 m m Module Lesson

11 Find the lateral area and surface area of the cylinder. Leave your answer in terms of π ft 4 ft 11 in. 7 in. Find the total surface area of the composite figure. Round to the nearest tenth ft 14 ft 6 ft 14 ft 8 ft 14 ft 14 ft 8 ft 1 ft Module Lesson

12 Find the total surface area of the composite figure. Round to the nearest tenth cm ft ft 0.5 ft cm 6 cm 9 cm ft 10 cm 1 ft 11. The greater the lateral area of a florescent light bulb, the more light the bulb produces. One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical bulb is 3 inches long with a 3 -inch radius. Which bulb will produce more light? 4 1. Find the lateral and surface area of a cube with edge length 9 inches. 13. Find the lateral and surface area of a cylinder with base area 64π m and a height 3 meters less than the radius. Module Lesson

13 14. Biology Plant cells are shaped approximately like a right rectangular prism. Each cell absorbs oxygen and nutrients through its surface. Which cell can be expected to absorb at a greater rate? (Hint: 1 μm = 1 micrometer = meter) 7 µm 35 µm 10 µm 15 µm 11 µm 15 µm ge07se_c10l04003a AB 15. Find the height of a right cylinder with surface area 160π f t and radius 5 ft. 16. Find the height of a right rectangular prism with surface area 86 m, length 10 m, and width 8 m. 17. Represent Real-World Problems If one gallon of paint covers 50 square feet, how many gallons of paint will be needed to cover the shed, not including the roof? If a gallon of paint costs $5, about how much will it cost to paint the walls of the shed? 1 ft 1 ft 18 ft 18 ft Module Lesson

14 18. Match the Surface Area with the appropriate coin in the table. Coin Diameter (mm) Thickness (mm) Surface Area (m m ) Penny Nickel Dime Quarter A B C D Algebra The lateral area of a right rectangular prism is 144 c m. Its length is three times its width, and its height is twice its width. Find its surface area. 0. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another cylinder that has a radius of 4 cm and the same surface area as the first cylinder. Module Lesson

15 H.O.T. Focus on Higher Order Thinking 1. Analyze Relationships Ingrid is building a shelter to protect her plants from freezing. She is planning to stretch plastic sheeting over the top and the ends of the frame. Assume that the triangles in the frame on the left are equilateral. Which of the frames shown will require more plastic? Explain how finding the surface area of these figures is different from finding the lateral surface area of a figure. 10 ft 10 ft 10 ft 10 ft 10 ft ge07sec10l04004aa 1st pass 4/3/5 cmurphy ge07sec10l04005a 1st pass 4/1/5 cmurphy. Communicate Mathematical Ideas Explain how to use the net of a three-dimensional figure to find its surface area. 3. Draw Conclusions Explain how the edge lengths of a rectangular prism can be changed so that the surface area is multiplied by 9. Module Lesson

16 Lesson Performance Task A manufacturer of number cubes has the bright idea of packaging them individually in cylindrical boxes. Each number cube measures inches on a side. 1. What is the surface area of each cube?. What is the surface area of the cylindrical box? Assume the cube fits snugly in the box and that the box includes a top. Use 3.14 for π. Module Lesson

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