Topic 8 of 9

Volume of rectangular prisms

Step 7 · Fraction and decimal arithmetic topics

Measure volume by counting unit cubes, find the volume of a box with length × width × height, and add the volumes of a shape made of two boxes.

Counting unit cubes

Volume is how much space a solid takes up. Measure it by counting the unit cubes that fill it with no gaps and no overlaps. A cube 1 centimeter on each edge is 1 cubic centimeter.

A box 4 cubes long, 3 wide and 2 high: 24 unit cubes.

Count by layers: the bottom layer is 4 × 3 = 12 cubes, and there are 2 layers, so 12 × 2 = 24 cubes.

Length × width × height

So the volume of a rectangular prism is length × width × height. Put another way: the area of the base times the height.

Example: Volume of a box

A box is 5 cm long, 4 cm wide and 3 cm high. What is its volume?

5 × 4 = 20 cubes in each layer.
3 layers: 20 × 3 = 60.

Answer60 cubic centimeters

Example: A missing edge

A box has a volume of 72 cubic inches. Its base is 6 inches by 4 inches. How tall is it?

The base holds 6 × 4 = 24 cubes.
72 ÷ 24 = 3 layers.

Answer3 inches

Adding volumes

A solid made of two boxes has a volume equal to the two volumes added together. Split it, find each volume, and add.

Box A holds 16 cubes and box B holds 12: 28 cubes together.

Example: Two boxes

A step is made of a box 6 m by 2 m by 1 m with a box 3 m by 2 m by 1 m on top. What is its volume?

6 × 2 × 1 = 12.
3 × 2 × 1 = 6.
12 + 6 = 18.

Answer18 cubic meters

Area is counted in squares and volume in cubes, so volume units are cubic: cubic centimeters (cm³) and cubic meters (m³).

Try it yourself

Work each one on paper first, then open the answer to check.

A box is 6 cm by 2 cm by 5 cm. What is its volume?
Show answer
60 cubic centimeters.
A cube has edges of 4 inches. What is its volume?
Show answer
64 cubic inches.
A box has a base of 10 square meters and a height of 3 meters. What is its volume?
Show answer
30 cubic meters.
A box has a volume of 48 cubic feet. It is 4 feet long and 3 feet wide. How high is it?
Show answer
4 feet.
Two boxes have volumes of 24 and 36 cubic centimeters. What is their total volume?
Show answer
60 cubic centimeters.