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**10-4 Surface Areas of Prisms and Cylinders Warm Up Problem of the Day**

Lesson Presentation Course 2

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Warm Up Find the volume of each figure to the nearest tenth. Use 3.14 for . 1. rectangular pyramid 7 ft by 8 ft by 10 ft tall 186.7 ft3 2. cone with radius 2 ft and height 3 ft 12.6 ft3 3. triangular pyramid with base area of 54 ft2 and height 9 ft 162 ft3

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Problem of the Day The volume of a 10-meter-tall square pyramid is 120 m3. What is the length of each side of the base? 6 m

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**Learn to find the surface area of prisms and cylinders.**

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Vocabulary net surface area

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If you remove the surface from a three-dimensional figure and lay it out flat, the pattern you make is called a net. Nets allow you to see all the surfaces of a solid at one time. You can use nets to help you find the surface area of a three-dimensional figure Surface area is the sum of the areas of all of the surfaces of a figure.

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**You can use nets to write formulas for the surface area of prisms**

You can use nets to write formulas for the surface area of prisms. The surface area S is the sum of the areas of the faces of the prism. For the rectangular prism shown, S = lw + lh + wh + lw + lh + wh = 2lw + 2lh + 2wh

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SURFACE AREA OF A PRISM The surface area of a rectangular prism is the sum of the areas of each face. S = 2lw + 2lh + 2wh

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**Additional Example 1: Finding the Surface Area of a Prism**

Find the surface area of the prism formed by the net. S = 2lw + 2lh + 2wh S = (2 · 15 · 9) + (2 · 15 · 7) + (2 · 9 · 7) Substitute. S = Multiply. S = 606 Add. The surface area of the prism is 606 in2.

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**Find the surface area of the prism formed by the net.**

Check It Out: Example 1 4 in. Find the surface area of the prism formed by the net. 6 in. 3 in. 3 in. 4 in. S = 2lw + 2lh + 2wh S = (2 · 4 · 6) + (2 · 4 · 3) + (2 · 6 · 3) Substitute. S = Multiply. S = 108 Add. The surface area of the prism is 108 in2.

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If you could remove the lateral surface from a cylinder, like peeling a label from a can, you would see that it has the shape of a rectangle when flattened out. You can draw a net for a cylinder by drawing the circular bases (like the ends of a can) and the rectangular lateral surface as shown below. The length of the rectangle is the circumference, 2r, of the cylinder. So the area of the lateral surface is 2rh. The area of each base is r2. r Circumference of cylinder (2r) h

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**SURFACE AREA OF A CYLINDER**

The surface area S of a cylinder is the sum of the areas of its bases, 2r2, plus the area of its lateral surface, 2rh. S= 2r2 + 2rh

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**Additional Example 2: Finding the Surface Area of a Cylinder**

Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for . 6 ft 8.3 ft 6 ft S = 2r2 + 2rh Use the formula. S (2 · 3.14 · 62) + (2 · 3.14 · 6 · 8.3) Substitute. S Multiply. S Add. S 538.8 Round. The surface area of the cylinder is about ft2.

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Check It Out: Example 2 Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for . 9 ft 20 ft 9 ft S = 2r2 + 2rh Use the formula. S (2 · 3.14 · 92) + (2 · 3.14 · 9 · 20) Substitute. S Multiply. S 1,639.08 Add. S 1,639.1 Round. The surface area of the cylinder is about 1,639.1 ft2.

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**Additional Example 3: Problem Solving Application**

What percent of the total surface area of the soup can is covered by the label? 2 in. Soup 6 in. 5 in. Course 2

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**Understand the Problem**

Additional Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Find the area of the label, and compare to the total surface area of the can. List the important information: • The can is approximately cylinder shaped. • The height for the can is 6 inches. • The radius of the can is 2 inches. • The height of the label is 5 inches. Course 2

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**Additional Example 3 Continued**

2 Make a Plan Find the surface area of the can and the area of the label. Divide to find the percent of the surface area covered by the label. Soup 6 in. 5 in. 2 in. Course 2

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**Additional Example 3 Continued**

Solve 3 S = 2r2 + 2rh 2(3.14)(2)2 + 2(3.14)(2)(6) Substitute for r and h. in.2 A = lw Substitute 2r for l . = (2r)w (2)(3.14)(2)(5) Substitute for r and w. 62.8 cm2 Percent of the surface covered by the label: = 62.5 %. About 62.5% of the can’s surface is covered by the label. 62.8 cm cm2 Course 2

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**Additional Example 3 Continued**

4 Look Back Estimate the areas of the two rectangles in the net. Label: 2(3)(2)(5) = 60 in2 Can: 2(3)(2)(6) = 77 in2 60 in.2 77 in.2 = 78% The answer should be less than 78% because you did not consider the area of the two circles. So 62.5% is reasonable. Course 2

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Check it Out: Example 3 What percent of the total surface area of the oil can is covered by the label? 6 in. 10 in. 4 in. Course 2

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**Understand the Problem**

Check It Out: Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Find the area of the label, and compare to the total surface area of the can. List the important information: • The can is approximately cylinder shaped. • The height for the can is 10 inches. • The diameter of the can is 6 inches. • The height of the label is 4 inches. Course 2

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**Check It Out: Example 3 Continued**

2 Make a Plan Find the surface area of the can and the area of the label. Divide to find the percent of the surface area covered by the label. 10 in. 4 in. 6 in. Course 2

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**Check It Out: Example 3 Continued**

Solve 3 S = 2r2 + 2rh Substitute for r and h. 2(3.14)(3)2 + 2(3.14)(3)(10) in2 A = lw Substitute 2r for l . = (2r)w (2)(3.14)(3)(4) Substitute for r and w. cm2 Percent of the surface covered by the label: = 30.7 %. About 30.7% of the can’s surface is covered by the label. 75.36 in in.2 Course 2

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**Check It Out: Example 3 Continued**

4 Look Back Estimate the areas of the two rectangles in the net. Label: 2(3)(3)(4) = 72 in2 Can: 2(3)(3)(10) = 180 in2 72 in in.2 = 40% The answer should be less than 40% because you did not consider the area of the two circles. So 30.7% is reasonable. Course 2

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Lesson Quiz Find the surface area of each figure to the nearest tenth. 1. 2. 100.5 ft2 352 ft2 3. A drum is cylindrical, and its 14 in. width fits into a drum stand. What percent of the total surface area of the drum is covered by the 3 in. red stripe? Use 3.14 for p. 25%

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