Math Core

Counting cubes layer by layer works, but real boxes don't come with cubes drawn on them. If you know a box's length, width, and height, you can multiply to find its volume.

From layers to a formula

A box shape with 6 rectangle faces is called a rectangular prism. Picture one that is 5 cm long, 3 cm wide, and 4 cm tall.

A rectangular prism 5 cm long, 3 cm wide, and 4 cm tall.

Imagine filling it with centimeter cubes.

The bottom layer has 5 cubes in each row and 3 rows: $5 \times 3 = 15$ cubes.
The prism is 4 cm tall, so it holds 4 layers.
In all: $15 \times 4 = 60$ cubes.

Look at what you multiplied: length, then width, then height. The volume is 60 cubic centimeters.

Volume of a rectangular prism

V = \text{length} \times \text{width} \times \text{height}

V = l \times w \times h

The answer is in cubic units.

Base times height

The bottom face of the prism is called the base. Its area, $B$ , tells you how many cubes fit in one layer. The height tells you how many layers there are. So you can also write:

V = B \times h

For the prism above, $B = 5 \times 3 = 15$ square cm and $h = 4$ cm, so $V = 15 \times 4 = 60$ cubic cm. It is the same answer, found the same way.

Any order works

You can multiply the three numbers in any order. Pick the order that makes the math easy. For a box 25 cm by 7 cm by 4 cm, do $25 \times 4 = 100$ first, then $100 \times 7 = 700$ .

Worked example: Use the formula

A cereal box is 8 inches long, 2 inches wide, and 12 inches tall. What is its volume?

V = 8 \times 2 \times 12 = 16 \times 12 = 192

The volume is 192 cubic inches.

Worked example: A cube

A cube has edges that are each 6 feet long. What is its volume?

All three dimensions are 6.

V = 6 \times 6 \times 6 = 36 \times 6 = 216

The volume is 216 cubic feet.

Worked example: Base area given

The base of a fish tank has an area of 200 square inches. The tank is 12 inches tall. How much water can it hold?

V = B \times h = 200 \times 12 = 2{,}400

The tank holds 2,400 cubic inches of water.

Worked example: A missing edge

A box has a volume of 72 cubic cm. It is 6 cm long and 4 cm wide. How tall is it?

The base holds $6 \times 4 = 24$ cubes in one layer. How many layers make 72?

24 \times ? = 72 \qquad 72 \div 24 = 3

The box is 3 cm tall.

Common mistake

Multiply all three dimensions. $5 \times 3 = 15$ gives only the bottom layer, not the whole box. And never add the dimensions: $5 + 3 + 4 = 12$ is not a volume.

Tip

Check your units. A length times a length times a length gives cubic units. If your answer says "square," you forgot the height.

Practice

Practice 1

A rectangular prism is 4 units long, 3 units wide, and 2 units tall. What is its volume, in cubic units?

Type a number.

Practice 2

Which expression gives the volume of a box that is 9 cm long, 5 cm wide, and 2 cm tall?

Practice 3

A cube has edges that are each 5 inches long. What is its volume, in cubic inches?

Type a number.

Practice 4

The base of a box has an area of 18 square cm. The box is 5 cm tall. What is its volume, in cubic cm?

Type a number.

Practice 5

A toy chest is 10 inches wide, 8 inches deep, and 6 inches tall. What is its volume, in cubic inches?

Type a number.

Practice 6

A swimming pool for a backyard is shaped like a rectangular prism. It is 20 feet long, 14 feet wide, and 5 feet deep. How many cubic feet of water fill it to the top?

Type a number.

Practice 7

A box has a volume of 84 cubic cm. It is 4 cm long and 3 cm wide. How tall is it, in cm?

Type a number.