Introduction

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Think It Through

Lesson 24 Understand Volume

Lesson 24Understand Volume

What does volume measure?

The volume of a three-dimensional figure is the amount of space contained inside the figure. Volume is measured by the number of cubic units that can be packed into a figure.

A cube with edge lengths of 1 unit is called a unit cube. A unit cube has 1 cubic unit of volume.

Area is the number of square units needed to cover a plane figure. Volume is the number of cubic units needed to fill a solid figure.

A plane figure is a two-dimensional figure. To find the area of a plane figure, you need measurements in two dimensions: length and width. One way you can find the area of a plane figure is by covering it in unit squares, without gaps or overlap. A unit square has an area of 1 square unit.

A solid figure is a three-dimensional figure. To find the volume of a solid figure, you need measurements in three dimensions: length, width, and height. One way you can find the volume of a solid figure is by packing it with unit cubes, without gaps or overlaps.

1 unit

1 unit

1 unit

Think How is volume different from area?

Circle the figures below that have volume.

Area 5 4 square units

Unit Cube

Volume 5 4 cubic units

Unit Square

5.MD.C.3a

5.MD.C.3b

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Suppose you need to buy a storage box to hold all of your number cubes. All of the cubes are the same size, with edges that are 1 unit long. How would you decide which storage box to buy?

One way to make sure you are buying a big enough box is to use volume. You need a storage box with a volume that will hold all of your cubes.

You know you have 24 number cubes. You can use that number and the volume of each cube to find the volume of the box you need. Then you could make sure that you buy a box big enough to hold all of the cubes.

Volume 5 24 cubic units

8 cubes ineach layer

Reflect1 Explain how 9 square units and 9 cubic units are different.

Think When might you need to measure volume?

I need to count how many cubes are in the whole box, but I can’t see all of the cubes. I can see that there are 8 cubes in 1 layer and that there are 3 of the same-size layers. 8 1 8 1 8 5 24

Guided Instruction

Think About

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Lesson 24

Lesson 24 Understand Volume

Finding Volume with Unit Cubes

2 The measure of each edge length of a unit cube is .

3 The volume of each unit cube is .

Alexander stacked unit cubes to build the rectangular prism below. Use the rectangular prism to answer problems 4 and 5.

4 There are unit cubes in the bottom layer and there are layers. The figure has unit cubes.

5 The volume of the figure is cubic units.

Now try these two problems. Use the figures below to answer problems 6 and 7. Each figure was built using unit cubes.

A

B

6 What is the volume of Figure A?

7 How many of Figure A does it take to fill or build Figure B?

What is the volume of Figure B?

Let’s Explore the Idea You can use unit cubes to build or fill a solid figure. The volume of the figure is the number of unit cubes needed to build or fill the solid figure.

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Let’s Talk About It Solve the problems below as a group.

8 Why do you use cubic units instead of square units to find the volumes of solid figures?

9 Candace places blocks with a volume of 1 cubic unit each into her jewelry box, as shown at the right.

How many blocks does it take to fill the jewelry box?

What is the volume of Candace’s jewelry box?

10 Candace has another jewelry box that is like the box in problem 9, but taller. It holds 1 more layer of blocks. What is the volume of this second box?

Try It Another Way Work with your group to show the connection between the volume of the solid figure below, the number of layers in the figure, and the number of unit cubes in each layer.

11 How many unit cubes are in each layer?

How many layers are in the figure?

12 Explain how you can use multiplication to find the volume of the figure. Then find the volume.

1 cubic unit

Guided Practice

Connect

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Lesson 24

Ideas About Volume with Unit Cubes

Lesson 24 Understand Volume

Talk through these problems as a class. Then write your answers below.

13 Infer Eli is stacking unit cubes in a box. He partially fills the box, pauses, and says, “The volume of this box is 18 cubic units.”

Explain how Eli found the volume of the box.

14 Explain Zoe says that a box that is 1 unit wide, 2 units long, and 3 units tall has a greater volume than a box that is 2 units wide, 3 units long, and 1 unit tall. Is she correct? Explain your answer.

15 Compare Each cube in Figures A and B has a volume of 1 cubic unit. Which figure has less volume, Figure A or Figure B? Explain your answer.

A

B

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Independent Practice

Apply

Lesson 24

Ideas About Volume with Unit Cubes

Lesson 24 Understand Volume

16 Put It Together Use what you have learned to complete this task.

Niles used 16 blocks, each having a volume of 1 cubic unit, to build a rectangular prism.

Part A Draw or build a model to represent the situation.

Part B Look at your model. Describe the number of layers and the number of blocks in each layer. Then describe a different rectangular prism that also has a volume of 16 cubic units.