Student Outcomes Students determine the surface area of three-dimensional figures, those that are composite figures and those that have missing sections. Lesson Notes This lesson is a continuation of Lesson 23. Students will continue to work on surface area advancing to figures with missing sections. Classwork MP.7 & MP.8 Example 1 (8 minutes) Students should solve this problem on their own. Example 1 Determine the surface area of the image. of top and bottom prisms: Lateral sides: Base face: Base face with hole: = 44(1111 iiii. 33 iiii. ) = 111111 iiii = 1111 iiii. 1111 iiii. = 111111 iiii = 1111 iiii. 1111 iiii. 44 iiii. 44 iiii. = 111111 iiii There are two of these, making up 888888 iiii. of middle prism: Lateral sides = 44(44 iiii. 88 iiii. ) = 111111 iiii Scaffolding: As in Lesson 23, students can draw nets of the figures to help them visualize the area of the faces. They could determine the area of these without the holes first and subtract the surface area of the holes. : 888888 iiii + 111111 iiii = 999999 iiii Describe the method you used to determine the surface area. Answers will vary: I determined the surface area of each prism separately and added them together. Then, I subtracted the area of the sections that were covered by another prism. If all three prisms were separate, would the sum of their surface areas be the same as the surface area you determined in this example? No, if the prisms were separate, there would be more surfaces shown. The three separate prisms would have a greater surface area than this example. The area would be greater by the area of four 4 in. 4 in. squares (64 in 2 ). 266
Example 2 (5 minutes) Example 2 a. Determine the surface area of the cube. SSSSSSSSSSSSSS aaaaaaaa = 66ss SSSS = 66(1111 iiii. ) SSSS = 66(111111 iiii ) SSSS = 888888 iiii Scaffolding: As in Lesson 23, students can draw nets of the figures to help them visualize the area of the faces. They could determine the area of these without the holes first and subtract the surface area of the holes. Explain how 6(12 in. ) 2 represents the surface area of the cube. The area of one face, one square with a side length of 12 in., is (12 in. ) 2, and so a total area of all six faces is 6(12 in. ) 2. b. A square hole with a side length of 44 inches is drilled through the cube. Determine the new surface area. Area of interior lateral sides: = 44(1111 iiii. 44 iiii. ) = 111111 iiii of cube with holes: = 66 (1111 iiii. ) (44 iiii. 44 iiii. ) + 44(1111 iiii. 44 iiii. ) = 888888 iiii 3333 iiii + 111111 iiii = 11111111 iiii How does drilling a hole in the cube change the surface area? We have to subtract the area of the square at the surface from each end. What happens to the surfaces that now show inside the cube? These are now part of the surface area. What is the shape of the piece that was removed from the cube? A rectangular prism was drilled out of the cube with the following dimensions: 4 in. 4 in. 12 in. How can we use this to help us determine the new total surface area? We can find the surface area of the cube and the surface area of the rectangular prism, but we will have to subtract the area of the square bases from the cube and also exclude these bases in the area of the rectangular prism. Why is the surface area larger when holes have been cut into the cube? There are more surfaces showing now. All of the surfaces need to be included in the surface area. 267
Explain how the expression 6(12 in. ) 2 2(4 in. 4 in. ) + 4(12 in. 4 in. ) represents the surface area of the cube with the hole. From the total surface area of a whole (uncut) cube, 6(12 in. ) 2, the area of the bases (the cuts made to the surface of the cube) are subtracted: 6(12 in. ) 2 2(4 in. 4 in. ). To this expression we add the area of the four lateral faces of the cutout prism, 4(12 in. 4 in. ). The complete expression then is 6(12 in. ) 2 2(4 in. 4 in. ) + 4(12 in. 4 in. ). Example 3 (5 minutes) Example 3 A right rectangular pyramid has a square base with a side length of 1111 inches. The surface area of the pyramid is iiii. Find the height of the four lateral triangular faces. Area of base Area of the four faces = 1111 iiii. 1111 iiii. = 111111 iiii = iiii 111111 iiii = 111111 iiii The total area of the four faces is 111111 iiii. Therefore, the area of each triangular face is 4444 iiii. Area of lateral side = 11 bbbb 4444 iiii = 11 (1111 iiii. )hh 4444 iinn = (55 iiii.)hh hh = 88 iiii. The height of each lateral triangular face is 88 inches. What strategies could you use to help you solve this problem? I could draw a picture of the pyramid and label the sides so that I can visualize what the problem is asking me to do. What information have we been given? How can we use the information? We know the total surface area, and we know the length of the sides of the square. We can use the length of the sides of the square to give us the area of the square base. How will the area of the base help us determine the slant height? First, we can subtract the area of the base from the total surface area in order to determine what is left for the lateral sides. Next, we can divide the remaining area by 4 to get the area of just one triangular face. Finally, we can work backward. We have the area of the triangle, and we know the base is 10 in., so we can solve for the height. 268
Exercises 1 8 (20 minutes) Students work in pairs to complete the exercises. Exercises Determine the surface area of each figure. Assume all faces are rectangles unless it is indicated otherwise. 1. Top and bottom Extra interior sides Left and right sides Front and back sides = (1111 mm 55 mm) = 111111 mm = (55 mm 77 mm) = 7777 mm = (55 mm 1111 mm) = 111111 mm = (1111 mm 1111 mm) (88 mm 77 mm) = ( mm 5555 mm ) = (111111 mm ) = 333333 mm = 111111 mm + 7777 mm + 111111 mm + 333333 mm = 666666 mm 2. In addition to your calculation, explain how the surface area of the following figure was determined. The surface area of the prism is found by taking the sum of the areas of the trapezoidal front and the areas of the back of the four different-sized rectangles that make up the lateral faces. Area top Area bottom Area sides = cccc 88 cccc = cccc = cccc 88 cccc = 111111 cccc = ( cccc 88 cccc) + (1111 cccc 88 cccc) = 333333 cccc Area front and back = 11 (1111 cccc + cccc)( cccc) = (444444 cccc ) = 999999 cccc = cccc + 111111 cccc + 333333 cccc + 999999 cccc = 11111111 cccc 269
3. of prisms on the sides: Area of front and back = ( iiii. 1111 iiii. ) = 5555 iiii Area of top and bottom = ( iiii. 1111 iiii. ) = 4444 iiii Area of side Area of side with hole = 1111 iiii. 1111 iiii. = 111111 iiii = 1111 iiii. 1111 iiii. 33 iiii. 33 iiii. = 111111 iiii There are two such rectangular prisms, so the surface area of both is 777777 iiii. of middle prism: Area of front and back = (33 iiii. 1111 iiii. ) = 7777 iiii = (33 iiii. 1111 iiii. ) = 7777 iiii = 777777 iiii + 111111 iiii = 888888 iiii 4. In addition to your calculation, explain how the surface area was determined. The surface area of the prism is found by taking the area of the base of the rectangular prism and the area of its four lateral faces and adding it to the area of the four lateral faces of the pyramid. Area of base = 99 ffff. 99 ffff. = 8888 ffff Area of rectangular sides = 44(99 ffff. 55 ffff. ) = 111111 ffff Area of triangular sides = 44 11 (99 ffff. )(66 ffff. ) = 111111 ffff = 8888 ffff + 111111 ffff + 111111 ffff = 333333 ffff 5. A hexagonal prism has the following base and has a height of uuuuuuuuuu. Determine the surface area of the prism. Area of bases Area of 55 unit sides Area of other sides = (4444 + 66 + + 1111) uuuuuuuuss = 111111 uuuuuuuuss = 44(55 88) uuuuuuuuss = 111111 uuuuuuuuss = (44 88) uuuuuuuuss + (1111 88) uuuuuuuuss = 111111 uuuuuuuuss = 111111 uuuuuuuuss + 111111 uuuuuuuuss + 111111 uuuuuuuuss = 444444 uuuuuuuuss 270
6. Determine the surface area of each figure. a. SSSS = 66ss = 66(99 mm) = 66(8888mm ) = 444444 mm b. A cube with a square hole with 33 mm side lengths has been drilled through the cube. Lateral sides of the hole of cube with holes = 44(99 mm 33 mm) = 111111 mm = 444444 mm (33 mm 33 mm) + 111111 mm = 555555 mm c. A second square hole with 33 mm side lengths has been drilled through the cube. SSSSSSSSSSSSSS aaaaaaaa = 555555 mm 44(33 mm 33 mm) + 44(33 mm 33 mm) = 666666 mm 7. The figure below shows cubes with an edge length of 11 unit. Determine the surface area. Area top and bottom = uuuuuuuuuu Area sides = 1111 uuuuuuuuuu Area front and back = uuuuuuuuuu = + 1111 + = 7777 uuuuuuuuuu 271
8. The base rectangle of a right rectangular prism is 44 ffff. 66 ffff. The surface area is ffff. Find the height. Let hh be the height in feet. Area of one base: 44 ffff. 66 ffff. = fftt Area of two bases: Numeric area of four lateral faces: Algebraic area of four lateral faces: ( fftt ) = 4444 fftt fftt 4444 fftt = fftt (66hh + 44hh) Solve for hh. (6666 + 4444) = 111111 = 111111 hh = 1111 The height is 1111 feet. Closing (2 minutes) Write down three tips that you would give a friend that is trying to calculate surface area. Lesson Summary To calculate the surface area of a composite figure, determine the surface area of each prism separately, and add them together. From the sum, subtract the area of the sections that were covered by another prism. To calculate the surface area with a missing section, find the total surface area of the whole figure. From the total surface area, subtract the area of the missing parts. Then, add the area of the lateral faces of the cutout prism. Exit Ticket (5 minutes) 272
Name Date Exit Ticket Determine the surface area of the right rectangular prism after the two square holes have been drilled. Explain how you determined the surface area. 273
Exit Ticket Sample Solutions Determine the surface area of the right rectangular prism after the two square holes have been drilled. Explain how you determined the surface area. Area of top and bottom Area of front and back Area inside = (1111 cccc 66 cccc) = 111111 ccmm = (66 cccc 88 cccc) = 9999 ccmm = (1111 cccc 88 cccc) 44(55 cccc 55 cccc) = 111111 ccmm = 88(55 cccc 66 cccc) = ccmm = 111111 ccmm + 9999 ccmm + 111111 ccmm + ccmm = 666666 ccmm Take the sum of the areas of the four lateral faces and the two bases of the main rectangular prism, and subtract the areas of the four square cuts from the area of the bases of the main rectangular prism. Finally, add the lateral faces of the prisms that were cut out of the main prism. Problem Set Sample Solutions Determine the surface area of each figure. 1. In addition to the calculation of the surface area, describe how you found the surface area. Area of top Area of bottom Area of left and right sides Area of front and back sides = cccc 1111 cccc = 333333 ccmm = cccc 1111 cccc = 333333 ccmm = cccc cccc + 1111 cccc cccc = 999999 ccmm = (1111 cccc 1111 cccc) + 11 (55 cccc 1111 cccc) = (111111 ccmm + 3333 ccmm ) = ( ccmm ) = 444444 ccmm = 333333 ccmm + 333333 ccmm + 999999 ccmm + 444444 ccmm = ccmm Split the area of the two trapezoidal bases into triangles and rectangles, take the sum of the areas, and then add the areas of the four different-sized rectangles that make up the lateral faces. 274
2. Area of front and back = (1111. 44 iiii. 88. 66 iiii. ) = 333333. 4444 iinn = (88. 66 iiii. iiii. ) = 444444. 88 iinn Area of top and bottom = (1111. 44 iiii. iiii. ) (44. 88 iiii.. iiii. ) = (444444. 66 iinn 1111. 5555 iinn ) = (444444. 0000 iinn ) = 888888. 0000 iinn = 333333. 4444 iinn + 444444. 88 iinn + 888888. 0000 iinn = 11111111. 3333 iinn 3. Area of front and back = 11 (3333 mm + 1111 mm)1111 mm = 777777 mm Area of top = 1111 mm 3333 mm = 555555 mm 32 m Area of left and right sides = (1111 mm 3333 mm) = (666666 mm ) = 11111111 mm Area of bottom = 3333 mm 3333 mm = 11111111 mm = 777777 mm + 11, 111111 mm + 11, mm + 555555 mm = 33333333 mm 4. Determine the surface area after two square holes with a side length of mm are drilled through the solid figure composed of two rectangular prisms. of top prism before the hole is drilled: Area of top = 44 mm 55 mm = mm Area of front and back = (44 mm 55 mm) = 4444 mm = (55 mm 55 mm) = 5555 mm of bottom prism before the hole is drilled: Area of top = 1111 mm 1111 mm mm = 8888 mm Area of bottom = 1111 mm 1111 mm = 111111 mm Area of front and back = (1111 mm 33 mm) = 6666 mm of interiors: Area of interiors = 44( mm 44 mm) + 44( mm 33 mm) = 5555 mm = (1111 mm 33 mm) = 6666 mm SSSSSSSSSSSSSS aaaaaaaa = 111111 mm + 333333 mm + 5555 mm 1111 mm = 444444 mm 275
5. The base of a right prism is shown below. Determine the surface area if the height of the prism is 1111 cccc. Explain how you determined the surface area. Take the sum of the areas of the two bases made up of two right triangles, and add to it the sum of the areas of the lateral faces made up of rectangles of different sizes. = ( cccc 1111 cccc) + (1111 cccc 1111 cccc) + ( cccc 1111 cccc) + (77 cccc 1111 cccc) = ccmm + 111111 ccmm + ccmm + 7777 ccmm = 666666 ccmm Area of bases = 11 (77 cccc cccc) + 11 ( cccc 1111 cccc) = (77 cccc cccc) + ( cccc 1111 cccc) = 111111 ccmm + 333333 ccmm = 444444 ccmm = 666666 ccmm + 444444 ccmm = 11111111 ccmm 276