The volume of a rectangular pyramid is defined as the three-dimensional space occupied by this figure. We can calculate the value of this volume by multiplying the area of the base by the height of the pyramid and dividing by three. This means that we need three lengths to calculate the volume of these pyramids, two lengths for the base and one length for the height. Since it is a three-dimensional measurement, the volume is measured in cubic units such as cm³, m³.

Here, we will learn about the formula that we can use to calculate the volume of a rectangular pyramid. In addition, we will solve some problems in which we will apply this formula.

## Formula to find the volume of a rectangular prism

The volume of any pyramid is calculated by multiplying the area of its base by the length of its height and dividing by 3. Therefore, we have the following formula:

$latex \text{Volume} = \frac{1}{3}\text{Area base} \times \text{Height}$

In a rectangular pyramid, its base is a rectangle. Remember that the area of a rectangle is calculated by multiplying the length of its base by the length of its width. Therefore, we have the following formula:

$latex V=\frac{1}{3}b\times a\times h$ |

where *b* is the length of the rectangular base, *a* is the width of the rectangular base, and *h* es the height of the pyramid.

## Volume of a rectangular pyramid – Examples with answers

The formula for the volume of a rectangular pyramid is used to solve the following examples. Each example has its respective solution, but it is recommended that you solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the volume of a pyramid that has a height of 5 m and a rectangular base with a length of 4 m and a width of 5 m?

##### Solution

We have the following lengths:

- Pyramid height, $latex h=5$
- Rectangle base, $latex b=4$
- Rectangle width, $latex a=5$

Using these lengths in the volume formula, we have:

$latex V=\frac{1}{3}b\times a\times h$

$latex V=\frac{1}{3}\times4\times 5\times 5$

$latex V=33.3$

The volume is equal to 33.3 m³.

**EXAMPLE 2**

If a rectangular pyramid has a base of 6 m, a width of 5 m, and a height of 6 m, what is its volume?

##### Solution

From the question, we get the following values:

- Pyramid height, $latex h=6$
- Rectangle base, $latex b=6$
- Rectangle width, $latex a=5$

Using these values in the volume formula, we have:

$latex V=\frac{1}{3}b\times a\times h$

$latex V=\frac{1}{3}\times6\times 5\times 6$

$latex V=60$

The volume is equal to 60 m³.

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**EXAMPLE 3**

A pyramid has a height of 7 m. If its rectangular base has a width of 6 m and a base of 8 m, what is its volume?

##### Solution

We have the following information:

- Pyramid height, $latex h=7$
- Rectangle base, $latex b=8$
- Rectangle width, $latex a=6$

We use the volume formula with these values:

$latex V=\frac{1}{3}b\times a\times h$

$latex V=\frac{1}{3}\times 8\times 6\times 7$

$latex V=112$

The volume is equal to 112 m³.

**EXAMPLE 4**

A rectangular pyramid has a volume of 10 m³. If its height is 5 m and its width is 3 m, what is the length of its base?

##### Solution

We have the following:

- Volume, $latex V=10$
- Pyramid height, $latex h=5$
- Rectangle width, $latex a=3$

In this case, we start with the volume and want to find the length of its base. Therefore, we use the volume formula and solve for *b*:

$latex V=\frac{1}{3}b\times a\times h$

$latex 10=\frac{1}{3}\times b\times 3\times 5$

$latex 30= b\times 3\times 5$

$latex 30= 15b$

$latex b=2$

The length of the base is 2 m.

**EXAMPLE 5**

What is the length of the base of a rectangular pyramid that has a volume of 144 m³, a width of 6 m, and a height of 9 m?

##### Solution

We have the following values:

- Volume, $latex V=144$
- Pyramid height, $latex h=9$
- Rectangle width, $latex a=6$

We use the values given in the volume formula and solve for *b*:

$latex V=\frac{1}{3}b\times a\times h$

$latex 144=\frac{1}{3}\times b\times 6\times 9$

$latex 432= b\times 6\times 9$

$latex 432= 54b$

$latex b=8$

The length of the base is 8 m.

## Volume of a rectangular pyramid – Practice problems

Use the formula for the volume of rectangular pyramids to solve the following practice problems. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about triangular pyramids? Take a look at these pages:

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