Using a Volume Formula for Rectangular Prisms | Grade 6 Math | LevelKite

Volume is the amount of space a solid shape takes up. In this lesson you will find the volume of rectangular prisms. A rectangular prism is a box-shaped solid. Its six flat sides, called faces, are all rectangles. A line segment where two faces meet is an edge.

Here are the facts about measuring volume that this lesson builds on.

A unit cube is a cube whose edges are each $1$ unit long.
One unit cube has a volume of $1$ cubic unit. If its edges are $1$ centimeter long, its volume is $1$ cubic centimeter. You write that as $1 \text{ cm}^3$ .
When unit cubes fill a solid exactly, with no gaps and no overlaps, the number of cubes is the volume in cubic units.

Counting cubes one by one is slow. For a rectangular prism, multiplication does the counting much faster.

Think about a rectangular prism that is $4$ units long, $3$ units wide and $2$ units high. Its edge lengths are whole numbers, so whole unit cubes fill it exactly. Count the cubes in layers.

The bottom layer has $3$ rows of $4$ cubes. That is $4 \times 3 = 12$ cubes.
The prism is $2$ units high, so it has $2$ layers of $12$ cubes.
All together, that is $12 \times 2 = 24$ cubes.

The volume is $24$ cubic units. You did not count one cube at a time. You multiplied the length, the width and the height: $4 \times 3 \times 2 = 24$ .

Counting unit cubes in layers

Length (units)	Width (units)	Height (units)	Cubes in one layer	Number of layers	Volume (cubic units)
$4$	$3$	$2$	$4 \times 3 = 12$	$2$	$12 \times 2 = 24$
$5$	$2$	$3$	$5 \times 2 = 10$	$3$	$10 \times 3 = 30$
$6$	$4$	$5$	$6 \times 4 = 24$	$5$	$24 \times 5 = 120$

Read across any row of the table. The number of cubes in one layer is the length times the width. The number of layers is the same as the height. So each volume is the length times the width times the height.

You can write this pattern once, using letters, and then use it for other rectangular prisms. A rule written with symbols like this is a formula.

Key concept

The volume formula for a rectangular prism

Let $l$ be the length, $w$ the width and $h$ the height of a rectangular prism. These are the lengths of the three edges that meet at one corner. Let $V$ be the volume.

When you measure $l$ , $w$ and $h$ in the same unit, the volume is all three multiplied together. The answer is in the matching cubic unit. For example, edge lengths in centimeters give a volume in cubic centimeters.

Why this works when the edge lengths are whole numbers: $l \times w$ counts the unit cubes in one layer, and $h$ counts the layers.

V = l \times w \times h

Worked example

Using three edge lengths

A rectangular prism is $6$ cm long, $4$ cm wide and $3$ cm high. What is its volume?

List what you know and check the units. The length is 6 cm, the width is 4 cm and the height is 3 cm. All three are in centimeters, so you can use the formula as it is.
$l = 6 \text{ cm},\quad w = 4 \text{ cm},\quad h = 3 \text{ cm}$
Write the formula. It fits because you know the length, the width and the height.
$V = l \times w \times h$
Replace each letter with its value. Each letter stands for one edge length of this prism, so the equation now describes this prism.
$V = 6 \times 4 \times 3$
Multiply the length by the width first. This counts the 1-centimeter cubes in one layer.
$V = (6 \times 4) \times 3 = 24 \times 3$
Multiply by the height, because the prism has 3 layers of 24 cubes.
$V = 24 \times 3 = 72$
Attach the unit. You multiplied three lengths in centimeters, so the volume is in cubic centimeters.
$V = 72 \text{ cm}^3$

Answer

The volume of the prism is $72$ cubic centimeters.

You can write the same formula in a second way. Choose one face of the prism to be the base. The height $h$ is then the length of an edge that meets the base at a right angle. It tells you how far the prism reaches from the base to the opposite face.

Let $B$ be the area of the base, in square units such as square meters ( $\text{m}^2$ ). If the base has length $l$ and width $w$ , then $B = l \times w$ . So you can put $B$ in place of $l \times w$ in the first formula. The volume does not change, because $B$ and $l \times w$ are the same number.

Here is the second form. Use it when the base area and the height are in matching units, such as square meters and meters. Those units give a volume in cubic meters.

V = B \times h

Worked example

Using a base area and a height

The base of a rectangular prism has an area of $35$ square meters. The height of the prism, measured at a right angle to that base, is $6$ meters. What is the volume of the prism?

List what you know and check the units. The base area is 35 square meters and the height is 6 meters. Square meters and meters match, so you can use the formula as it is.
$B = 35 \text{ m}^2,\quad h = 6 \text{ m}$
Pick the form that uses the base area. The length and the width are not given on their own, and you do not need them.
$V = B \times h$
Substitute the values. The base area already stands for length times width, so the height is the only number left to multiply by.
$V = 35 \times 6$
Multiply. Split 35 into 30 and 5, which does not change its value. Multiply each part by 6 and add the results.
$V = (30 \times 6) + (5 \times 6) = 180 + 30 = 210$
Attach the unit. Square meters times meters gives cubic meters.
$V = 210 \text{ m}^3$

Answer

The volume of the prism is $210$ cubic meters.

Worked example

Finding the space inside a moving van

The cargo space of a moving van is shaped like a rectangular prism. It is $15$ feet long, $7$ feet wide and $6$ feet high. How many cubic feet of space does it have?

All three edge lengths are in feet, so you can write the formula and substitute the values.
$V = l \times w \times h = 15 \times 7 \times 6$
Change the order of the numbers so that an easy pair comes first. You may do this because changing the order of the numbers you multiply does not change the result. The volume stays the same.
$V = 15 \times 7 \times 6 = 15 \times 6 \times 7$
Multiply the first two numbers. They are the easy pair you moved together.
$V = (15 \times 6) \times 7 = 90 \times 7$
Multiply by the last number. You multiplied three lengths in feet, so the unit is cubic feet.
$V = 90 \times 7 = 630 \text{ ft}^3$

Answer

The cargo space has a volume of $630$ cubic feet.

Takeaway. The volume of a rectangular prism tells you how many cubic units fill it. When you know three edge lengths in the same unit, multiply them: $V = l \times w \times h$ . When you know the base area and the height in matching units, multiply those: $V = B \times h$ . The two forms agree, because a base that is $l$ long and $w$ wide has area $B = l \times w$ . Write every volume in cubic units.

Every edge length in this lesson was a whole number. Next, you will find the volume of prisms whose edge lengths are fractions.