? LESSON 10.1 ESSENTIAL QUESTION Surface Area of Prisms How do you find the surface area of a prism? Expressions, equations, and relationships make connections to the formulas for lateral and total surface area and determine solutions for problems involving prisms EXPLORE ACTIVITY Modeling the Surface Area of a Prism The surface area of a three-dimensional figure is the sum of the areas of all of its surfaces. You can use a net to help you explore formulas related to the surface area of a prism. A net for a CD storage box made of balsa wood is shown. A B C The front and have the same area. A = lh = = square inches The left side and have the same area. A = wh = = square inches The lateral faces of a prism are parallelograms that connect the bases. The lateral area (L) of a prism is the sum of the areas of its lateral faces. Find the lateral area of the box. side 12 in. 12 in. top back base front side L = 2 + 2 = square inches D Find the perimeter (P) of the base of the box. E F G P = + + + = inches Multiply the perimeter of the base by the height. Ph = = square inches What do you notice about your answers in parts C and E? The top and have the same base area (B). B = lw = = square inches Math Talk Mathematical Processes Write a formula for total surface area S in terms of l, w, and h. H Add the lateral area to the area of the two bases to find the total surface area (S). S = 2B + L = 2 + = square inches Lesson 10.1 269
Surface Area of a Rectangular Prism Lateral and Total Surface Area of a Prism Math On the Spot The lateral area L of a prism is the perimeter P of the base times the height h of the prism. P h L = Ph B The total surface area S of a prism is twice the area of a base B plus the lateral area L. P S = 2B + L or S = 2B + Ph h B EXAMPLE 1 Problem Solving Joanna is wrapping a present in the box shown. Find the amount of wrapping paper in square inches that Joanna needs, not counting overlap. Analyze Information 1 ft 8 in. Math Talk Mathematical Processes Is it possible to wrap the box with a 30-inch by 20-inch piece of wrapping paper? Explain. 270 Animated Math Unit 3 What are you asked to find? You need to find the total surface area of the box. List the important information. The length of the box is 1 foot. The width of the box is 8 inches. The height of the box is 6 inches. Formulate a Plan Change 1 foot to 12 inches. Then find the area and perimeter of a base of the box, and substitute into the formula for the surface area of a prism. Solve Area of Base: B = l w = 12 8 = 96 Perimeter of Base: P = 2l + 2w = 2(12) + 2(8) = 40 S = 2B + Ph Total surface area of a prism S = 2(96) + (40)(6) S = 192 + 240 = 432 Substitute: B = 96, P = 40, and h = 6. Simplify. The total surface area of the box is 432 square inches.
Justify and Evaluate You can also find the area of the front, top, and side, and multiply each by 2 to find the surface area. Front: 12 6 = 72 Top: 12 8 = 96 Side: 8 6 = 48 2 72 = 144 2 96 = 192 2 48 = 96 S = 144 + 192 + 96 = 432 square inches, so the answer is correct. YOUR TURN 1. Sara is lining the bottom and lateral faces of a drawer with liner paper. The dimensions of the inside of the drawer are 1 yard, 20 inches, and 9 inches. What is the total area in square inches being covered? Personal Math Trainer Online Assessment and Intervention Surface Area of a Triangular Prism When you find the total surface area of a triangular prism, don t confuse the base and height of the triangular base with the base and height of the prism. EXAMPLE 2 Math On the Spot Kevin is painting a block in the shape of a triangular prism. What is the surface area of the block? 3 in. STEP 1 STEP 2 STEP 3 Find the area of a base. B = 1 2 bh B = 1 (8)(3) = 12 2 Find the perimeter of a base. P = 8 + 5 + 5 = 18 Area of a triangle Substitute. Use the formula for surface area. 8 in. Perimeter of a triangle S = 2B + Ph S = 2(12) + (18)(6) Substitute: B = 12, P = 18, and h = 6 S = 24 + 108 = 132 The surface area of the triangular prism is 132 in 2. Lesson 10.1 271
YOUR TURN Personal Math Trainer Online Assessment and Intervention Find the lateral area and total surface area of each prism. 2. 10 cm 3. 6 cm 10 cm 8 cm 6. 7 in. 6. Guided Practice Find the lateral area and total surface area of each prism. (Example 1) 1. 2. 9 in. 5 ft 2 yd 4 ft 3. Akira plans to cover the box shown in contact paper. Find the amount of contact paper that Akira needs, not counting overlap. (Example 1) 9 in. 7 in.? 4. A gift box is in the shape of a triangular prism. How much cardboard is needed to construct the box, not counting overlap? (Example 2) ESSENTIAL QUESTION CHECK-IN 2 cm 2.5 cm 5. What are two ways that you can find the surface area of a prism? 6.3 cm 5.2 cm 3 cm 272 Unit 3
Name Class Date 10.1Independent Practice Personal Math Trainer Online Assessment and Intervention Find the lateral and total surface area of each prism. Round to the nearest tenth if necessary. 6. 7. 8. 2 ft 4.5 ft 18 in. 9 cm 4 ft 1 ft 6 cm 80 mm 9. 4 ft 3 ft 10. 11. 5 ft 7.5 cm 8.5 cm 8.5 cm 6 ft 5 cm 8 cm 1 1 ft 1 ft 1 13 in. 12. The bases of a prism are right triangles with side lengths 6 meters, 8 meters, and 10 meters. The height of the prism is 3 meters. What is the lateral area of the prism? What is the total surface area? 13. A rectangular prism has a length of 8 inches and a width of 7 inches. The lateral area is 150 square inches. What is the height of the prism? 14. Multiple Representations Write a formula for the total surface area of a cube in terms of its edge length x. Explain your reasoning. Lesson 10.1 273
15. Multistep Matt bought a tent without a floor. Estimate the surface area of the tent in square feet. 87 in. 16. Multistep Keeshawn is building a box with a lid out of plywood with the dimensions shown. Plywood costs $0.50 per square foot. Find the cost of the plywood Keeshawn needs for the box. 63 in. 87 in. 4 yd 4 ft 10 ft 17. A glass prism on a chandelier is 75 millimeters long. A base of the prism is an equilateral triangle with side lengths of 9 millimeters and a height of about 7.8 millimeters. What is the approximate surface area of the prism? FOCUS ON HIGHER ORDER THINKING 18. Problem Solving A cube with an edge length of 4 inches is painted on all of its sides. Then the cube is cut into 64 cubes with an edge length of 1 inch. What percent of the total surface area is painted? Explain. 2 1 yd Work Area 19. Problem Solving The base of a triangular prism is a right triangle whose legs are 7 cm and 24 cm. The height of the prism is 30 cm. What is the lateral area of the prism? Explain how you found your answer. 20. Communicate Mathematical Ideas Explain how to find the surface area of the composite figure. Then find its surface area. 1 in. 1.8 in. 1.8 in. 3 in. 3 in. 274 Unit 3