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LESSON 5.2 Nets and Solids A net is a flat figure that can be folded to form a closed, three-dimensional object. This type of object is called a geometric solid . Vocabulary net solid Materials scissors graph paper Which of these figures are also nets for a cube? That is, which will fold into a cube? Cut out a copy of each figure. Try to fold it into a cube. Find and draw three other nets for a cube. With your class, compile all the different nets for a cube that were found. How many nets are there? Do you think your class found all of them? Explore Inv 1 Use a Net 229 Inv 2 Use Nets to Investigate Solids 230 Inv 3 Is Today’s Beverage Can the Best Shape? 233 228 CHAPTER 5 Geometry in Three Dimensions 228_239_C05_L2_888704_SE_C2.indd228 228 228_239_C05_L2_888704_SE_C2.indd228 228 5/27/08 11:40:46 AM 5/27/08 11:40:46 AM

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  • L E S S O N

    5.2Nets and Solids

    A net is a flat figure that can be folded to form a closed, three- dimensional object. This type of object is called a geometric solid .

    Vocabulary net

    solid

    Materials scissors

    graph paper

    Which of these figures are also nets for a cube? That is, which will fold into a cube? Cut out a copy of each figure. Try to fold it into a cube.

    Find and draw three other nets for a cube. With your class, compile all the different nets for a cube that were found. How many nets are there? Do you think your class found all of them?

    Explore

    Inv 1 Use a Net 229

    Inv 2 Use Nets to Investigate Solids 230

    Inv 3 Is Todays Beverage Can the Best Shape? 233

    228 CHAPTER 5 Geometry in Three Dimensions

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  • Use a Net

    For a net to form a three-dimensional object, certain lengths have to match. For example, in the nets for cubes, the side lengths of all the squares must be the same. You will now investigate whether other nets will form closed solids. Pay close attention to the measurements that need to match.

    Decide whether each figure is a net, that is, whether it will fold into a solid. One way to decide is to cut out a copy of the figure and try to fold it. If the figure is a net, describe the shape it creates. If it is not a net, explain why not.

    1. 2.

    3. 4.

    5. 6.

    Materials scissors

    Investigation 1Materials scissors

    Investigation 1

    Develop & Understand: ADevelop & Understand: A

    Lesson 5.2 Nets and Solids 229

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  • 7. 8.

    9.

    1. For a net to fold into a cylinder, the circumference of the circles must be the same as what other length?

    2. Draw a net for a figure different from those in Exercises 19. Explain how you know your net will fold to form a solid.

    Use Nets to Investigate Solids

    You have calculated surface area by counting the square units on the outside of a structure. If the faces of a solid are not squares, like the figures below, you can find the solids surface area by adding the areas of all its faces.

    Share & SummarizeShare & Summarize

    Materials scissors

    metric ruler

    Investigation 2Materials scissors

    metric ruler

    Investigation 2

    230 CHAPTER 5 Geometry in Three Dimensions

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  • This net folds to form a cylinder.

    10 cm

    5 cm

    Because the net folds into a cylinder, the circles must be the same size. The area of each circle is r 2 cm 2 , or 25 cm 2 . This is about 78.5 cm 2 .

    The length of the rectangle must be equal to the circumference of each circle, which is 10 cm, or about 31.4 cm. So, the area of the rectangle is about 314 cm 2 .

    To find the cylinders surface area, add the three areas.

    78.5 cm 2 + 78.5 cm 2 + 314 cm 2 = 471 cm 2

    Each net shown will fold to form a closed solid. Use the nets measurements to find the surface area and volume of the solid. If you cannot find the exact volume, approximate it. It may be helpful to cut out and fold a copy of the net.

    1. 2.

    2 cm

    1 cm

    3 m

    3 m

    6 m

    ExampleExample

    Develop & Understand: ADevelop & Understand: A

    Math LinkThe formula for the circumference of a circle is 2r, where r is the radius; or d, where d is the diameter.

    The formula for the area of a circle is r 2 .

    Lesson 5.2 Nets and Solids 231

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  • 3.

    2 m

    1 m

    4 m

    1.5 m

    4.

    3 m

    3 m

    4 m

    5 m

    5.

    2 cm

    2 cm4 cm

    Math LinkThe formula for the area A of a triangle is A = 1 _ 2 bh, where b is the base and h is the height.

    232 CHAPTER 5 Geometry in Three Dimensions

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  • 1. Explain how to find the surface area of a solid from the net for that solid.

    2. Here is a net for a rectangular solid. Take whatever measurements you think are necessary to find the solids volume and surface area. Explain what measurements you took and how you used them.

    Is Todays Beverage Can the Best Shape?

    The Bursting Bubbles company is trying to reduce its manufacturing costs in order to increase its profits. The president of the company wonders if there is a way to use less material to make a beverage can.

    In this investigation, you will help Bursting Bubbles design a can that has the minimum surface area for the volume of beverage it contains.

    To get started, inspect a 12-ounce beverage can. Get an idea of its dimensions, volume, and surface area.

    1. Find the area in square centimeters of the base of the can. If the can you are using is indented at the bottom, assume that it is not.

    2. Measure the cans height in centimeters.

    Share & SummarizeShare & Summarize

    Materials 12-oz beverage can

    metric ruler

    Investigation 3Materials 12-oz beverage can

    metric ruler

    Investigation 3

    Develop & Understand: ADevelop & Understand: A

    Lesson 5.2 Nets and Solids 233

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  • The volume printed on the can is given in milliliters (abbreviated mL). A milliliter has the same volume as a cubic centimeter, so 1 mL = 1 cm 3 .

    3. Does multiplying the area of the base by the height give the volume that is printed on the can? If not, why might the measures differ?

    4. What is the surface area of the can in square centimeters? How did you find it? (Hint: Imagine what a net for the can would look like.)

    Now consider how the can design might be changed. For these exercises, use the volume that you calculated in Exercises 14, rather than the volume printed on the can.

    5. Design and describe five other cans that have approximately the same volume as your can. Sketch a net for each can. Record the radius of the base and the height. Include cans that are both shorter and taller than a regular beverage can. (Hint: First choose the height of the can and then find the radius.)

    6. Calculate the surface area of each can you designed. Record your groups data on the board for the class to see.

    7. Compare the surface areas of the cans that you and your classmates found. Which can has the greatest surface area? Which has the least surface area?

    Bursting Bubbles wants to make a can using the least amount of aluminum possible. The company cannot do it by evaluating every possible can, like the five you found, and choosing the best. There are too many, an infinite number, and Bursting Bubbles could never be sure it had found the one using the least material.

    One strategy that mathematicians use is to gather data, as you did in Exercise 5, and look for patterns in the data. For an exercise like this one, they would also try to show that a particular solution is the best in a way that does not involve testing every case.

    8. Use what you know about volume and any patterns that you found in Exercises 5 and 6 to recommend a can with the least possible surface area to the president of Bursting Bubbles. Describe your can completely. Explain why you believe it has the least surface area.

    Does the shape of standard beverage cans use the minimum surface area for the volume they contain? If not, what might be some reasons companies use the shape they do?

    Develop & Understand: BDevelop & Understand: B

    Share & SummarizeShare & Summarize

    Real-World LinkAs water bubbles up through the earth from natural springs, it collects minerals and carbon dioxide. In the 1700s, chemists perfected a method of adding carbonation and minerals to plain water and bottling the result.

    234 CHAPTER 5 Geometry in Three Dimensions

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  • On Your Own ExercisesLesson 5.2

    Decide whether each figure is a net, that is, whether it will fold into a closed solid. One way to decide is to cut out a copy of the figure and try to fold it. If the figure is a net, describe the shape it creates. If it is not a net, explain why not.

    1. 2.

    3.

    4. A square pyramid has a square base and triangular faces that meet at a vertex. Draw at least three nets for a square pyramid.

    In Exercises 58, find the surface area of the solid that can be created from the net. Show how you found the surface area.

    5. Each triangle is equilateral with sides 5 centimeters and height 4.3 centimeters.

    6. The radius of each circle is 2 centimeters. The height of the parallelogram is 2 centimeters.

    Practice & ApplyPractice & Apply

    Lesson 5.2 Nets and Solids 235

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  • On Your Own Exercises

    7. The pentagons have side lengths of 3 centimeters. If you divide each pentagon into five isosceles triangles, the height of each triangle is 2 centimeters. The other shapes are squares.