Chapter 11 · Volume
Measure and Compare Volumes in Cubic Units
Count unit cubes to name volumes in cubic centimeters, cubic inches and cubic feet. Choose a sensible unit and compare volumes by checking the units first.
You already know how to find a volume. You count the unit cubes that fill a solid with no gaps and no overlaps.
But a count leaves one question open: how big is each cube? Twelve tiny cubes fit in one hand. Twelve very large cubes might not fit in your room. Both counts are , yet the two spaces are nothing alike.
In this lesson, you will use unit cubes with an exact size. That size gives every volume a unit you can name.
This lesson uses three unit cubes whose sizes everyone agrees on. The length of each cube's edges gives its unit a name. (The edges of a cube are the straight lines where its flat faces meet. They are all the same length.)
Picture a cube with edges centimeter long. The space it fills is 1 cubic centimeter. In the same way, a cube with edges inch long fills 1 cubic inch, and a cube with edges foot long fills 1 cubic foot.
So the name of the unit tells you which cube to picture. When you read cubic inch, think of a cube that is inch long, inch wide and inch tall. To keep things short, this lesson calls the three cubes the centimeter cube, the inch cube and the foot cube.
| Cubic unit | Its unit cube | Short form | About as big as |
|---|---|---|---|
cubic centimeter | edges centimeter long | cubic cm | the smallest cube in a set of base-ten blocks |
cubic inch | edges inch long | cubic in | a small ice cube |
cubic foot | edges foot long | cubic ft | a box that a basketball fits inside |
Worked example
Name a volume by counting centimeter cubes
Jo builds a solid from cubes. Each cube has edges centimeter long. The solid has layers, and each layer has cubes. What is the volume of the solid?
Find the size of one cube before you count. Its edges are 1 centimeter long, so one cube fills 1 cubic centimeter. That is the unit for this volume.
Count the cubes one layer at a time, so you count every cube exactly once. Write one 6 for each of the 3 layers, then add.
Join the count to the unit. The solid is made of 18 cubes and each cube fills 1 cubic centimeter, so the solid fills 18 cubic centimeters.
Answer
The volume of the solid is cubic centimeters, which you can also write as cubic cm.
Now change only the cube. Build Jo's solid again with inch cubes, using the same layers of . Its volume is cubic inches, or cubic in. Build it with foot cubes, and its volume is cubic feet, or cubic ft.
The count is each time, but the three solids are not the same size. The centimeter cube is the smallest cube and the foot cube is the largest. So cubic cm sits in your hand, while cubic ft is a stack of boxes that could each hold a basketball.
That is why you always say the unit with the number. It is also why you pick a sensible unit before you measure: a cube that fits in the space and gives a count you can manage.
| Object | A sensible unit | Why it is sensible |
|---|---|---|
A small box of paper clips | cubic centimeters | The box is small, so it needs a small cube. Centimeter cubes fit inside it. |
A cereal box | cubic inches | Inch cubes fit inside. A foot cube is too big to fit. |
A closet | cubic feet | The closet is large. Foot cubes fit inside, and you do not need a huge number of them. |
What happens to the count when you swap the cube? Think about filling one space two times.
A small cube fills only a little of the space, so it takes more small cubes to fill it. A large cube fills a lot of the space, so it takes fewer large cubes to fill it. The space has not changed. Only the size of the cube has changed.
Smaller cube, bigger count. Larger cube, smaller count.
Worked example
Predict the count and choose a unit
Exactly cubes with edges foot long fill a toy chest. Kai wants to fill the same chest with cubes that have edges inch long. Will Kai need more than cubes or fewer than ? Which unit is more sensible for the chest, cubic feet or cubic inches?
Name the volume from the foot cubes. A cube with edges 1 foot long fills 1 cubic foot, and 8 of these cubes fill the chest.
Compare the two cubes. 1 foot is as long as 12 inches, so 12 of the inch cubes fit in a single row along one edge of a foot cube. An inch cube is much smaller than a foot cube.
Decide more or fewer. Each inch cube fills less space, so it takes more of them to fill the same chest. Just one row inside one foot cube already uses 12 of them, and 12 is more than 8.
Choose the unit. Foot cubes fit in the chest and give a small count that is easy to picture. Inch cubes also fit, but their count would be huge and hard to work with.
Answer
Kai will need many more than cubes, because each inch cube fills less space than a foot cube. Cubic feet is the more sensible unit, and the chest has a volume of cubic feet.
Key concept
A volume is a number and a cubic unit together
Every volume has two parts. The number tells you how many unit cubes fill the space. The cubic unit tells you how big each cube is: edges of centimeter, inch or foot.
You need both parts to know how much space there is. If you measure the same space with a smaller cube, the number goes up, but the space stays the same.
Now you are ready to compare volumes. Read the units first.
When two volumes have the same cubic unit, every cube you count is the same size. More cubes means more space, so you can compare the numbers. For example, cubic cm is larger than cubic cm because , and both numbers count centimeter cubes.
When the units are different, the two numbers count cubes of different sizes. Then the numbers alone cannot tell you which volume is larger.
Worked example
Compare three boxes
Box A holds layers with cubes in each layer. Box B holds layers with cubes in each layer. Every one of these cubes has edges inch long, and the cubes fill each box exactly. Box C has a volume of cubic ft. Which is larger, Box A or Box B, and by how much? Dana says Box C is the smallest because is the smallest number. Is Dana right?
Name the volumes of Box A and Box B. Each cube has edges 1 inch long, so each cube fills 1 cubic inch. Count layer by layer: four 9s for Box A and five 8s for Box B. Box A is 36 cubic inches and Box B is 40 cubic inches.
Check the units. Both volumes are in cubic inches, so both numbers count cubes of the same size. That lets you compare the numbers.
Subtract to find how much larger Box B is. Both numbers count inch cubes, so the difference is in cubic inches too.
Check the unit for Box C. It is cubic feet, so its 2 counts foot cubes, not inch cubes. You cannot rank Box C by comparing 2 with 36 and 40.
Think about the size of one foot cube. 1 foot is as long as 12 inches, so its bottom layer holds 12 rows, and each row has 12 of the inch cubes. That one layer already holds more inch cubes than all of Box B, so Box C is larger than both boxes.
Answer
Box B is larger than Box A by cubic inches. Dana is not right. Box C is the largest box, because the bottom layer of just one foot cube holds of the inch cubes, which is more than Box B holds.
Here is the idea to keep: a volume is a number and a cubic unit together. The number counts the unit cubes. The unit tells you whether each cube has edges of centimeter, inch or foot.
- To name a volume, count the cubes and say the unit of the cube you counted.
- Pick a cube that suits the object. A smaller cube always gives a bigger count for the same space.
- Compare the numbers only when the cubic units match.