Math Core

In 5th grade you found the volume of a box by counting unit cubes or multiplying length, width and height. Real boxes are rarely whole numbers of inches, though. A cereal box might be $7\frac{1}{2}$ inches wide. In this lesson you will see that the same volume formula works when the edges are fractions.

Volume is counting cubes

Volume is the amount of space inside a solid. It is measured in cubic units: how many unit cubes (cubes 1 unit on each edge) it takes to fill the solid.

When an edge length is a fraction, use smaller cubes. A cube with edges of $\frac{1}{2}$ inch is a lot smaller than a 1-inch cube. How much smaller? Along each edge of a 1-inch cube, 2 small cubes fit. So a 1-inch cube holds

2 \times 2 \times 2 = 8

of the half-inch cubes. That means each half-inch cube has a volume of $\dfrac{1}{8}$ cubic inch. The formula agrees: $\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{8}$ .

Worked example: Packing with small cubes

A box is $1\frac{1}{2}$ inches long, 1 inch wide and $2\frac{1}{2}$ inches tall. How many $\frac{1}{2}$ -inch cubes fill it? What is its volume?

Count how many half-inch cubes fit along each edge:

Length: $1\frac{1}{2}$ inches holds 3 halves.
Width: 1 inch holds 2 halves.
Height: $2\frac{1}{2}$ inches holds 5 halves.

So the box holds $3 \times 2 \times 5 = 30$ small cubes. Each one is $\frac{1}{8}$ cubic inch, so the volume is

30 \times \frac{1}{8} = \frac{30}{8} = 3\frac{3}{4} \text{ cubic inches.}

Now try the formula directly: $\dfrac{3}{2} \times 1 \times \dfrac{5}{2} = \dfrac{15}{4} = 3\dfrac{3}{4}$ . Same answer!

The volume formula

Packing with small cubes always gives the same answer as multiplying the edge lengths. So you can skip the cubes and use the formula.

Volume of a rectangular prism

V = l \times w \times h \qquad \text{or} \qquad V = B \times h

$l$ , $w$ and $h$ are the length, width and height. $B$ is the area of the base ( $B = l \times w$ ). These work for fractions and decimals, not just whole numbers.

A box 3 1/2 inches long, 1 1/2 inches wide (deep) and 2 inches tall.

Worked example: Using the formula

Find the volume of the box in the picture.

Change the mixed numbers to fractions, then multiply:

V = 3\frac{1}{2} \times 1\frac{1}{2} \times 2 = \frac{7}{2} \times \frac{3}{2} \times \frac{2}{1} = \frac{42}{4} = 10\frac{1}{2}

The volume is $10\frac{1}{2}$ cubic inches.

Common mistake

Don't multiply the whole numbers and the fractions separately. $3\frac{1}{2} \times 1\frac{1}{2}$ is not $3 \times 1$ plus $\frac{1}{2} \times \frac{1}{2}$ . Always change mixed numbers to improper fractions (or decimals) before you multiply.

Worked example: A small cube

A cube has edges that are $\frac{2}{3}$ foot long. What is its volume?

All three edges of a cube are equal:

V = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}

The volume is $\dfrac{8}{27}$ cubic foot. That is less than 1, which makes sense because the cube is smaller than a 1-foot cube.

Finding a missing edge

If you know the volume and some of the edges, divide to find the missing one.

Worked example: A missing height

A planter box holds 30 cubic feet of soil. Its base is a rectangle with an area of $7\frac{1}{2}$ square feet. How tall is it?

Use $V = B \times h$ : $30 = 7\frac{1}{2} \times h$ .

h = 30 \div 7\frac{1}{2} = 30 \div \frac{15}{2} = 30 \times \frac{2}{15} = 4

The planter is 4 feet tall. Check: $7\frac{1}{2} \times 4 = 30$ . ✓

Tip

Look for easy pairs before multiplying. In $\frac{7}{2} \times \frac{3}{2} \times 2$ , the $\frac{1}{2}$ and the $2$ cancel, leaving $\frac{7}{2} \times 3 = \frac{21}{2}$ . Multiplying in a smart order keeps the numbers small.