lesson nets and solids - · pdf file5.2lesson nets and solids ... does the shape of standard...

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 Today’s 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.


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 1–9. 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 solid’s surface area by adding the areas of all its faces.

Share & SummarizeShare & Summarize


• metric ruler

Investigation 2Materials• scissors

• metric ruler

Investigation 2



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 πr 2 cm 2 , or 25π 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 10π cm, or about 31.4 cm. So, the area of the rectangle is about 314 cm 2 .

To find the cylinder’s 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 net’s 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


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

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



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.



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 solid’s volume and surface area. Explain what measurements you took and how you used them.

Is Today’s 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 can’s height in centimeters.


Materials• 12-oz beverage can

• metric ruler

Investigation 3Materials• 12-oz beverage can

• metric ruler

Investigation 3




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 1–4, 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 group’s 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


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.



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 5–8, 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



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.

8. Each triangle is equilateral with sides 8 centimeters and height 7 centimeters. The other shapes are squares.

9. A box of tissues has length 24 centimeters, width 12 centimeters, and height 10.5 centimeters.

a. What is the volume of the box?

b. What is the surface area of the box?

10. Alana has a juice pitcher that is shaped like a cylinder. It is 30 centimeters high and has a base with radius 6 centimeters. It has a flat lid.

a. How much juice will her pitcher hold?

b. What is the surface area of her pitcher and its lid?




11. Challenge This net will fold into a triangular pyramid. Find the surface area of the pyramid. Estimate its volume. You may want to fold a copy of the net into the pyramid.

5.2 cm

6 cm

12. Choose a container in your house. You might choose a rectangular box, a can other than a beverage can, or some other object.

a. Draw a net for the object.

b. Find the surface area and volume of the object by measuring its dimensions.

c. Draw nets for three other objects with the same volume as your container. Find the surface area of each. The new objects do not have to be real containers from your home.

d. Is there a prism or cylinder with less surface area but the same volume as your container? If so, draw a net for it.

13. Imagine removing a label from a soup can.

a. What is the shape of the label?

b. If the radius of the soup can is r and the height is h, what are the dimensions of the label?

SOUP

Create a net for each block structure.

14.

15.

16.

Connect & ExtendConnect & Extend




17. Challenge A cone has a circular base and a vertex some height away from that base. Make a net for a cone.

18. Challenge A tetrahedron is a solid with four triangular faces. You have made tetrahedrons from nets in this lesson. Nets for a tetrahedron contain only triangles. Can you make a net for another figure using only triangles? (Hint: You might try taping paper triangles together to form a new solid.)

19. Kinu’s Frozen Yogurt store packs its product into cylindrical tubs that are 20 cm in diameter and 30 cm tall.

a. How much frozen yogurt does a tub hold?

b. Kinu’s sells frozen yogurt in cylindrical scoops. If each scoop has base radius 2 cm and height 6 cm, what is the volume of a single scoop?

c. How many scoops of frozen yogurt are in a tub? Show how you found your answer.

d. What is the surface area of a tub?

e. What is the total surface area of the frozen yogurt from one tub after it has all been scooped?

f. Which has more surface area, the frozen yogurt in the tub or the scooped frozen yogurt? Which would melt faster?

20. In Your Own Words Explain why nets might be useful in sewing, manufacturing, or some other profession.




21. A circular pond is 100 feet in diameter. In the middle of winter, the ice on the pond is 6 inches thick.

a. Think of the ice on the pond as a cylinder. What is the volume of ice?

b. What is the surface area of the floating cylinder of ice?

c. What would be the edge length of a cube of ice that had the same volume?

d. What would be the surface area of a cube of ice that had the same volume?

e. Which has greater surface area, the cylinder of ice or the cube of ice? How much more?

Preview Match the following items with the most appropriate unit of weight.

22. an elephant A. ounces

23. a cup of yogurt B. pounds

24. a human being C. tons

Find each product or quotient.

25. 6(-12) 26. - 1 _ 5 (- 2

_ 3 )

27. 15 ÷ (-2) 28. - 4

_ 9 ÷ (- 2

_ 3 )

29. Copy and complete the chart.

Fraction 1 _ 2 3 _ 5

Decimal 0.5 0.65

Percent 50% 21%

Rewrite each number in standard notation.

30. 8 × 1 0 2 31. 1.42 × 1 0 6

32. 3.12 × 1 0 -4 33. 6.13 × 1 0 -2

34. Suppose you use a photocopying machine to reduce a photograph.

a. Your photograph is 10 cm wide, and you reduce it to 85% of its original size. What is the width of the reduced image?

b. You use the same setting to reduce the copy. What is the width of the image after the second reduction?

Mixed ReviewMixed Review



lesson nets and solids - · pdf file5.2lesson nets and solids ... does the shape of standard...

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