Volume and Surface Area of Rectangular Prisms All Boxed Up
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1 Volume and Surface Area of Rectangular Prisms SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Use Manipulatives, Visualization, Group Presentation, Think/Pair/Share The shape in Figure is a net. A net is a two-dimensional drawing used to represent or form a three-dimensional object or solid. Figure ACADEMIC VOCABULARY net ACTIVITY 4.. Cut out Figure on page 49 and fold it along the dotted lines to form a box. The figure formed is a cube. Describe the characteristics that make it a cube.. The net below will also form a cube. If face number is the bottom of the cube, which numbered face is the top of the cube? College Board. All rights reserved. 3. Many other nets can be used to represent a cube. a. Use graph paper to draw as many of these other nets as you can find. Cut out each net and fold it to verify that a cube can be formed. b. Sketch the nets you found that form a cube in the My Notes space. c. Sketch below two nets made up of six squares that do not form a cube. Unit 4 Three-Dimensional Geometry 4
2 ACTIVITY 4. Volume and Surface Area of Rectangular Prisms ACADEMIC VOCABULARY The area of a plane figure is the measure in square units of its interior region. SUGGESTED LEARNING STRATEGIES: : Interactive Word Wall, Summarize/Paraphrase/Retell, Discussion Group, Look for a Pattern In Elaine s new business,, she sells shipping materials, including boxes and packing peanuts. Her box supplier charges her cent per square inch of surface area for each box. Elaine can determine the surface area of a box by adding the areas of the six faces of the box (front, back, top, bottom, left, and right). Elaine must now find the surface areas of many boxes of different sizes, so she wants to find a pattern that will help shorten this work. Help her find the pattern. 4. One type of box that Elaine will keep in stock at is a cube. For one of the cubeshaped boxes, the length of each edge is 5 inches. 5 inches a. Find the area of each face. b. Find the total surface area of the cube. 5. Elaine wants to make a data table for the cube-shaped boxes. a. Record your answer from Item 4. Then complete the table for other cube-shaped boxes that will be in stock. Length of Edge (in.) Number of Faces Area of One Face (in. ) 00 Surface Area (in. ) College Board. All rights reserved. b. Describe any patterns you see in the table above. c. For each box, how does the area of one face relate to the surface area? 4 SpringBoard Mathematics with Meaning TM Level
3 Volume and Surface Area of Rectangular Prisms ACTIVITY 4. SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation, Identify a Subtask, Interactive Word Wall, Use Manipulatives 6. You can use a variable to represent the length of the edge of a cube. a. In the figure to the right, what variable is used to represent the length of the edge of the cube? e b. Write a rule, in terms of the variable, for finding the surface area of a cube. 7. Cube-shaped boxes with -inch edges are kept in stock at All Boxed Up. a. Determine the surface area of a box with -inch edges. b. The supplier charges Elaine cent per square inch of surface area for each box. Determine how much profit Elaine will make on a -inch cube-shaped box if she sells it for $4.95. Show your work. 00 College Board. All rights reserved. 8. Another type of box is a rectangular prism. In a rectangular prism, all the faces are rectangles but not necessarily squares. Look at Figure, which is a rectangular prism, on page 49. Show the calculations to find the area of each face. Then find the surface area of the prism. Face Calculation Area (in. ) MATH TERMS A prism is a solid with parallel congruent bases which are both polygons. The faces (sides) of a prism are all parallelograms or rectangles. A prism is named according to the shape of its bases Surface area of Figure = Unit 4 Three-Dimensional Geometry 43
4 ACTIVITY 4. Volume and Surface Area of Rectangular Prisms SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Think/Pair/Share 9. Cut out Figure on page 49. Fold it to form a rectangular prism. Make folds so that the measurements are on the outside. a. Label the length, width, and height of the rectangular prism you formed on this diagram. Face 6 Face 4 h = e Face w = l = b. Faces, 3, and 5 cannot be seen in the diagram above. Describe the location of each of these hidden faces. Face : Face 3: Face 5: c. Which pairs of faces have the same area? 0. Calculate the surface area of each rectangular prism. Show your work. a. b. inches 7 inches 5 inches 6 inches 00 College Board. All rights reserved. 30 inches 6 inches 44 SpringBoard Mathematics with Meaning TM Level
5 Volume and Surface Area of Rectangular Prisms ACTIVITY 4. SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, RAFT. Write a rule that Elaine can use to determine the surface area of a rectangular prism with the length, width, and height represented by the variables l, w, and h. Explain how you found the rule. h l w. Elaine is stocking a new box that is 4 inches long, inches wide, and 0 inches high. Make a recommendation to Elaine regarding a price for this size box. Remember that her box supplier charges her cent per square inch of surface area for each box. Be certain to explain how you arrived at your recommendation. 00 College Board. All rights reserved. Elaine plans to sell packing peanuts at. She knows that surface area will not tell her how many packing peanuts she needs to fill a box. She knows the volume of a box will help in estimating the number of packing peanuts needed to fill the box. Volume is measured in cubic units. Cubes are named for the lengths of their edges. A -inch cube is a cube with edges that are inch in length. A -inch cube is a cube with edges that are inches in length. Any size cubes can be used to build larger cubes. ACADEMIC VOCABULARY Volume is the measure of the space occupied by a solid. It is measured in cubic units such as cubic inches (in. 3 ). Unit 4 Three-Dimensional Geometry 45
6 ACTIVITY 4. Volume and Surface Area of Rectangular Prisms SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Look for a Pattern, Create Representations 3. Elaine uses unit cubes, similar to the ones your teacher has given you, as models of -inch cubes. a. Use your unit cubes to build models of -inch cubes and 3-inch cubes. Then complete this table. Length of Edge (in.) 3 Area of Face (in. ) Volume of Cube (in. 3 ) CONNECT TO AP The process of gathering numerical information, making and testing conjectures, and generalizing the information into formulas is a key component of all math courses. b. Describe any relationships you see in the table. 4. Describe how you can determine the volume of a cube when you do not have enough cubes to build the box. 5. Use the variable e to represent the length of the edge of a cube. Write a rule, in terms of e, for calculating the volume of a cube. 6. Determine the volume of a cube with edge length 8 inches. 7. Now think about boxes that are rectangular prisms. a. Complete this table. Use cubes to build boxes as needed. Length (in.) Width (in.) Height (in.) Volume (in. 3 ) College Board. All rights reserved. 46 SpringBoard Mathematics with Meaning TM Level
7 Volume and Surface Area of Rectangular Prisms ACTIVITY 4. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations, Identify a Subtask, RAFT b. Describe any pattern you see in the table for finding the volume of rectangular prisms (on the previous page). c. Use the variables l, w, and h to represent the length, width, and height of a box. Write a rule, in terms of l, w, and h, for finding the volume of a box. 8. Elaine has a customer who needs a box with a volume of cubic inches. The customer wants to know what size box is the least expensive to buy. The price will be based on surface area. The supplier charges cent per square inch. The dimensions of the box are whole numbers. What size box do you recommend that Elaine s customer buy? Write a report to Elaine that shows your work and explains your recommendation. 00 College Board. All rights reserved. Unit 4 Three-Dimensional Geometry 47
8 ACTIVITY 4. Volume and Surface Area of Rectangular Prisms CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work.. Determine whether the net can be cut out and folded to form a cube. 6. Danny needs to buy sand for this box. He wants to nearly fill the box, leaving only 6 inches empty at the top. How much sand does Danny need? 3 feet. What is the surface area of a cube that measures 4 cm on each edge? 3. Determine the length of one edge of a cube that has a surface area of 864 square inches. 4. Find the surface area of this prism. feet 7. A hole shaped like a rectangular prism is 3 feet wide, 5 feet long, and 3 feet deep. If the hole is made feet deeper, how much will the volume of the hole increase? 8. What is the maximum number of cubes with a side length of inches that can fit in this box? 6 inches 4 inches feet 0 cm 4 cm 5 cm 5. A swimming pool, in the shape of a rectangular prism, is 50 meters long, 5 meters wide and 3 meters deep. A liter is the same as 0.00 cubic meters. How many liters of water are needed to completely fill the pool? inches 9. Determine the length of a rectangular prism whose width is 5 mm and whose height is 8 mm, if the volume of the prism is 540 m m MATHEMATICAL You are to design a box REFLECTION that will hold 4 -inch cubes. Describe two possible designs for the box if there is no empty space left in the box after you have put in all of the cubes. Sketch each of your designs and give their surface areas and volumes. 00 College Board. All rights reserved. 48 SpringBoard Mathematics with Meaning TM Level
9 Volume and Surface Area of Rectangular Prisms ACTIVITY 4. Figure Figure 00 College Board. All rights reserved. inches inches Face inches inches 4 inches Face 4 inches Face 3 inches inches Face 4 inches 4 inches Face 5 4 inches inches Face 6 Unit 4 Three-Dimensional Geometry 49
10 00 College Board. All rights reserved.
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