Volume Formulas for Rectangular Prisms | Grade 5 Math | LevelKite

You have already found how many cubes fill a box shape by writing a multiplication expression. In this lesson you turn that idea into two formulas. A formula is a rule written with letters. It tells you how to find one number from other numbers.

A right rectangular prism is a box shape. It has 6 flat sides called faces, and every face is a rectangle. The faces meet at square corners. The line where two faces meet is called an edge. A shoebox and a brick are shaped like this.

Volume is the amount of space a solid shape fills. You measure it in cubic units. One cubic unit is the volume of a cube whose edges are each 1 unit long. A volume of 24 cubic units means the shape fills the same amount of space as 24 of those cubes.

Picture a right rectangular prism built from cubes that are each 1 unit long on every edge. It is 4 cubes long, 3 cubes wide and 2 cubes high.

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

All together, $4 \times 3 \times 2 = 24$ . The volume is 24 cubic units.

The three numbers you multiplied are the prism's edge lengths. They tell how long its edges are in each direction. The length is how long it is. The width is how wide it is. The height is how tall it is. Write $l$ for the length, $w$ for the width, $h$ for the height and $V$ for the volume. Then the same steps become a formula:

V = l \times w \times h

Worked example

Multiply the three edge lengths

A right rectangular prism is 8 centimeters long, 5 centimeters wide and 3 centimeters high. Find its volume.

Name the three edge lengths. They are all in centimeters, so the formula is ready to use.
$l = 8, \quad w = 5, \quad h = 3$
Write the formula. Put each number in place of its letter.
$V = l \times w \times h = 8 \times 5 \times 3$
Multiply the length by the width. This counts the 1-centimeter cubes in the bottom layer.
$8 \times 5 = 40$
Multiply by the height, because the prism has 3 layers of 40 cubes.
$40 \times 3 = 120$
Give the unit. Each cube is 1 centimeter long on every edge, so the volume is in cubic centimeters.
$V = 120 \text{ cubic centimeters}$

Answer

The volume of the prism is $120$ cubic centimeters.

There is a second way to write the same idea. In this lesson, the base of a prism is the face it sits on. The area of the base is the number of unit squares that cover that face. A unit square is a square whose sides are each 1 unit long, so area is measured in square units. Use the letter $B$ for the area of the base.

The base is a rectangle, so its area is the length times the width: $B = l \times w$ . In the prism you pictured, $B = 4 \times 3 = 12$ square units.

Now set one cube on each of those 12 squares. That makes the bottom layer. So the base area also tells you how many cubes are in one layer. The height tells you how many layers there are. This gives the second formula: $V = B \times h$ .

The two formulas agree because $B$ is $l \times w$ . In $V = l \times w \times h$ , you multiply the length by the width first, and that gives $B$ . After that, both formulas multiply by $h$ . The table shows both formulas at work on the same prism.

The same prism in cubes, in words and in symbols

In the prism	In words	In symbols	Value
Cubes in one row of the bottom layer	length	$l$	$4$ units
Rows in the bottom layer	width	$w$	$3$ units
Squares that cover the base	area of the base	$B = l \times w$	$4 \times 3 = 12$ square units
Layers of cubes	height	$h$	$2$ units
All the cubes	volume, from the three edge lengths	$V = l \times w \times h$	$4 \times 3 \times 2 = 24$ cubic units
All the cubes	volume, from the base area and the height	$V = B \times h$	$12 \times 2 = 24$ cubic units

Key concept

Two formulas, one volume

Think of a right rectangular prism as equal layers of cubes. The area of the base, $B$ , tells you how many cubes are in one layer. The height, $h$ , tells you how many layers there are. So the volume is $B \times h$ . Because $B$ is the length times the width, multiplying the three edge lengths gives the same volume. Measure every edge in the same unit, and state the volume in cubic units.

V = l \times w \times h = B \times h

Worked example

Multiply the base area by the height

The base of a right rectangular prism has an area of 28 square meters. The prism is 6 meters high. Find its volume.

Name what you know: the area of the base and the height. You do not know the length or the width. You do not need them, because the base area is already the length times the width.
$B = 28, \quad h = 6$
Write the formula that uses the base area. Put each number in place of its letter. This works because one layer holds 28 cubes and there are 6 layers.
$V = B \times h = 28 \times 6$
Multiply. Split 28 into 20 and 8 to make it easier. Since 20 times 6 is 120 and 8 times 6 is 48, add the two parts.
$28 \times 6 = 120 + 48 = 168$
Give the unit. The base area is in square meters and the height is in meters, so the volume is in cubic meters.
$V = 168 \text{ cubic meters}$

Answer

The volume of the prism is $168$ cubic meters.

When you check volume work, yours or someone else's, ask three questions.

Did the work multiply? Both formulas multiply. They never add the edge lengths.
Did it use every measurement? $V = l \times w \times h$ needs all three edge lengths. $V = B \times h$ needs the base area and the height.
Is the answer in cubic units? Square units measure the flat area of a face. Volume needs cubic units.

If an answer is no, you have found a mistake. Fix that part, then work the problem again.

Worked example

Find and fix the mistakes

A right rectangular prism is 6 feet long, 2 feet wide and 3 feet high. Theo writes $V = 6 + 2 + 3 = 11$ square feet. Find Theo's mistakes and give the correct volume.

Ask the first question: did the work multiply? No. Theo added the edge lengths. Adding puts three edges end to end and gives a length of 11 feet. It does not count cubes, so it is not the volume.
$6 + 2 + 3 = 11$
Fix the first mistake. Theo was right to use all three edge lengths, but they must be multiplied. The bottom layer has 6 times 2 cubes, and there are 3 layers.
$V = 6 \times 2 \times 3$
Work it out. First, 6 times 2 is 12. Then 12 times 3 is 36.
$6 \times 2 \times 3 = 12 \times 3 = 36$
Ask the last question: is the answer in cubic units? No. Theo wrote square feet, which measure flat area. Volume counts cubes that are 1 foot long on every edge, so the unit is cubic feet.
$V = 36 \text{ cubic feet}$

Answer

Theo made two mistakes: he added the edge lengths instead of multiplying them, and he used square feet. The correct volume is $36$ cubic feet.

Here is the big idea. The volume of a right rectangular prism is the area of its base times its height, and the area of the base is the length times the width. So $V = B \times h$ and $V = l \times w \times h$ are two ways to write the same count of cubes. When you find a volume, multiply, use every measurement the formula needs, and write the answer in cubic units.

Next, you will use these formulas to solve word problems and to find a missing edge length when you know the volume.