Surface Area of Solids
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1 Lesson 24 Surface Area of Solids Name: Prerequisite: Use a Net to Find Surface Area Study the example showing how to use a net to find the surface area of a prism. Then solve problems 7. Example Kioshi needs to find the surface area of a triangular prism with the dimensions shown. He begins by drawing a net of the triangular prism and labeling the dimensions of each face. Find the area of each face of the prism. 6 in. You can make a table that shows the dimensions and the area of each face of the prism. 20 in. 20 in. 20 in. 20 in. 20 in. 6 in. 6 in. Face Base (in.) Height (in.) Area (sq in.) Triangle Triangle Rectangle Rectangle Rectangle Why do the two triangles have the same base length and height? How did you find the area of each triangle? 2 Do all the rectangles have equal areas? Explain. 3 What is the surface area of the prism? Explain how you found the surface area. Vocabulary net a flat representation of a solid when it is unfolded. surface area the sum of the areas of all of the faces of a threedimensional figure. Curriculum Associates, LLC Copying is not permitted. Lesson 24 Surface Area of Solids 25
2 Solve. 4 A doorstop is a triangular prism with the dimensions shown. Label the side lengths (in centimeters) of the net for the doorstop. Then, inside each part of the net, write the area of that part in square centimeters. 3 cm 5 cm 2 cm 3 cm 5 Find the surface area of the prism in problem 4. Explain how you can find the surface area of any prism. 6 Write a product that represents the surface area of any rectangular prism with length l, width w, and height h. Explain. 7 The height of a rectangular prism is 20 centimeters. It has a surface area of 2,400 square centimeters. What are two possible sets of lengths and widths? Find one set of dimensions with l and w equal in length as well as a set of dimensions that are not equal. 252 Lesson 24 Surface Area of Solids Curriculum Associates, LLC Copying is not permitted.
3 Lesson 24 Name: Surface Area of a Triangular Prism Study the example problem showing how to find the surface area of a triangular prism. Then solve problems 7. Example The bases of the prism shown are isosceles triangles. What is the surface area of the prism? You can draw the faces of the prism, label the dimensions, and find the area of each face. There are two identical triangular faces. Draw and label one of them and find its area. 0.8 m.0 m.0 m A 5 (.2)(0.8) m.0 m 2 m.2 m.2 m Draw and label one of the identical rectangular faces. Then draw the third rectangular face. Find the areas..0 m.2 m 2 m 2 m A 5 (.0)(2) 5 2 A 5 (.2)(2) Surface area 5 2(0.48) 2(2) square meters What formula was used to find the area of the triangular faces? 2 What formula was used to find the area of the rectangular faces? 3 Why is the area of one of the rectangles different from the areas of the other two rectangles? Curriculum Associates, LLC Copying is not permitted. Lesson 24 Surface Area of Solids 253
4 Solve. 4 A manufacturer packages a product in a box shaped like a triangular prism with a height of 5.0 centimeters. The dimensions of the bases of the prism are shown. Draw each rectangular face of the box. Label the dimensions. Write the area of each rectangle inside it. 6.4 cm 8.0 cm 4.8 cm 5 Find the surface area of the box in problem 4. 6 If the height of the box in problem 4 is doubled from 5 centimeters to 0 centimeters, does the surface area double? Explain your result. 7 Find an expression for the surface area of a cube with edges of length s. Justify your answer. 254 Lesson 24 Surface Area of Solids Curriculum Associates, LLC Copying is not permitted.
5 Lesson 24 Name: Surface Area of a Complex Solid Study the example showing how to find the surface area of a complex solid. Then solve problems 5. Example Tiana has a storage box with the dimensions shown. To find the surface area, she draws and labels the faces of the prism as shown. Make a table with the area of each face. 26 in. 27 in. 26 in. A 27 in. 26 in. 27 in. C B 27 in. 26 in. D E F You can use the formulas you know to find the area of each figure that Tiana drew. Notice that figures A and B are composed of a triangle and a rectangle. Face Area (sq in.) Face Area (sq in.) A A 5 (3)(0) (0)(24) D A 5 (36)(26) B A 5 (3)(0) + (0)(24) E A 5 (36)(3) 5 08 C A 5 (36)(27) F A 5 (36)(0) What is the surface area of the box? 2 Chris says that he can find the surface area faster by thinking of the box as a triangular prism on top of a rectangular prism. He finds the surface area of each one and adds them together. Does his method make sense? Curriculum Associates, LLC Copying is not permitted. Lesson 24 Surface Area of Solids 255
6 Solve. 3 Milo is making a set of blocks for his child-care business. He has a block that is a triangular prism. He cuts the top off of this block to make a smaller triangular prism, which he paints red. A diagram of the original triangular face is shown. What is the surface area that Milo paints red? Original Block Cut 4 cm 4 cm 0 cm 5 cm 6 cm 5 cm 2 cm 2 cm 4 Refer to problem 3. Milo paints the lower piece green. Draw one of the trapezoidal faces and label its dimensions. What is the surface area of the block that Milo paints green? 4 cm 2 cm 6 cm 5 cm 0 cm 5 A metal machine part is 2 millimeters long, 2 millimeters wide, and 4 millimeters thick. A square that is 6 millimeters on each side is punched from the middle of the metal solid. What is the surface area of the resulting figure? 2 mm 4 mm 2 mm 256 Lesson 24 Surface Area of Solids Curriculum Associates, LLC Copying is not permitted.
7 Lesson 24 Name: Surface Area of Solids Solve the problems. Viola says that she can find the surface area of a rectangular prism by finding the area of one face and multiplying by 6. When is this true? When is this not true? Use diagrams to explain your answer. Think about the dimensions of a rectangular prism. 2 Which expression can be used to find the surface area of the triangular prism represented by the net? Select all that apply. A 3(0 5) (6 4) B 2(0 5) (0 6) (6 4) C (8 6) 2 4(5 4) (3 4) D (8 6) 2 2(5 4) (3 4) Which parts have equal areas? 5 in. 4 in. 5 in. 8 in. Curriculum Associates, LLC Copying is not permitted. Lesson 24 Surface Area of Solids 257
8 Solve. 3 What is the surface area of the prism shown? A.5 square meters B.8 square meters C 2.6 square meters D 2.4 square meters 0.5 m.2 m.3 m 0.4 m How many faces does this prism have? 4 A storage box without a top is 8 inches long, 5 inches wide, and 0 inches high. The outside of the box is painted yellow. What is the total area that is painted yellow? Explain. Would drawing a diagram help? 5 Lyle sliced a block of cheese along a diagonal into two triangular prisms, as shown. Lyle says that the surface area of the block is 2 times the surface area of one of the triangular prisms. Do you agree? Explain your answer. 4 in. 5 in. 4.5 in. How can you compare the surface areas? 6 Refer to problem 5. Find the difference between the surface area of the two triangular prisms and the surface area of the rectangular prism. What operation does the word difference indicate? 258 Lesson 24 Surface Area of Solids Curriculum Associates, LLC Copying is not permitted.
The surface area of a solid figure is the sum of the areas of its surfaces. To help you see all the surfaces of a solid figure, you can use a net.
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