Math Core

How much sand fits in a sandbox? How much cereal fits in a box? These questions ask about the space inside a solid shape. That amount of space is called volume.

Length, area, and now volume

You already measure in two ways.

Length measures in one direction. You use a ruler and count units like inches or centimeters.
Area covers a flat shape. You count unit squares.

Volume fills a solid shape, like a box. To measure it, you count unit cubes.

Definition

Unit cube

A unit cube is a cube whose edges are each 1 unit long. It has a volume of 1 cubic unit.

A unit cube. Every edge is 1 unit long.

Counting cubes

Here is a box packed with unit cubes. There are no gaps between the cubes, and no cubes overlap.

A box 3 cubes long, 2 cubes wide, and 2 cubes tall.

You can't see every cube, so counting one by one is tricky. Instead, count by layers.

The bottom layer has 3 cubes in each row and 2 rows. That is $3 \times 2 = 6$ cubes.
There are 2 layers, one on top of the other.
So the box holds $6 + 6 = 12$ cubes, or $2 \times 6 = 12$ .

The volume of the box is 12 cubic units.

Volume

Volume is the amount of space inside a solid. To find it, count how many unit cubes fill the solid with no gaps and no overlaps.

Units for volume

A unit cube can be any size. Its edge tells you its name.

Edge of the cube	Name of the unit	Short way to write it
1 centimeter	cubic centimeter	cm³
1 inch	cubic inch	in³
1 foot	cubic foot	ft³
1 meter	cubic meter	m³

Small things, like a gift box, are measured in cubic centimeters or cubic inches. Big spaces, like a room, are measured in cubic feet or cubic meters.

Worked example: Count by layers

A box is filled with centimeter cubes. The bottom layer is 4 cubes long and 3 cubes wide. There are 2 layers. What is the volume?

Bottom layer: $4 \times 3 = 12$ cubes.

Two layers: $12 + 12 = 24$ cubes.

The volume is 24 cubic centimeters.

Worked example: More layers

A stack of inch cubes has 5 cubes in each row and 2 rows in each layer. The stack is 3 layers tall. What is the volume?

Each layer: $5 \times 2 = 10$ cubes.

Three layers: $10 + 10 + 10 = 30$ cubes.

The volume is 30 cubic inches.

Worked example: Same volume, different shape

Solid A is 6 cubes long, 1 cube wide, and 1 cube tall. Solid B is 3 cubes long, 2 cubes wide, and 1 cube tall. Which has more volume?

Solid A: one layer of $6 \times 1 = 6$ cubes.

Solid B: one layer of $3 \times 2 = 6$ cubes.

They have the same volume, 6 cubic units, even though they look different.

Common mistake

Don't count only the cubes you can see. Some cubes are hidden in the back and in the middle. Count one layer, then count how many layers there are.

Tip

Volume uses cubic units. Area uses square units. If your answer is about filling a box, the unit should say "cubic."

Practice

Practice 1

What is the volume of one unit cube, in cubic units?

Type a number.

Practice 2

Which unit would you use to measure the volume of a shoebox?

Practice 3

A box has one layer of unit cubes. The layer is 5 cubes long and 3 cubes wide. What is the volume of the box, in cubic units?

Type a number.

Practice 4

Look at the box below. It is packed with unit cubes. What is its volume, in cubic units?

A box packed with unit cubes.

Type a number.

Practice 5

A tower is built from inch cubes. Each layer has 8 cubes. The tower has 4 layers. What is its volume, in cubic inches?

Type a number.

Practice 6

Maya fills a box with centimeter cubes. The bottom layer is 3 cubes long and 3 cubes wide. She makes 4 layers to fill the box. What is the volume of the box, in cubic centimeters?

Type a number.

Practice 7

Solid P is 2 cubes long, 2 cubes wide, and 2 cubes tall. Solid Q is 4 cubes long, 1 cube wide, and 1 cube tall. Which statement is true?