# Volume of a Square Pyramid – Formulas and Examples

The volume of a square pyramid defines the total space occupied by the pyramid in three-dimensional space. The measure of this volume is obtained by multiplying the area of the base of the pyramid by its height and dividing the product by three. The base is a square, so we find its area by squaring one of its side lengths. Volume is a three-dimensional measure, so we use cubic units to represent it.

Here, we will learn about the formula that we can use to calculate the volume of square pyramids. Then, we will use this formula to solve some practice problems.

##### GEOMETRY

Relevant for

Learning about the volume of a square pyramid.

See examples

##### GEOMETRY

Relevant for

Learning about the volume of a square pyramid.

See examples

## Formula to find the volume of a square prism

The volume of any pyramid is found by multiplying the area of its base times the length of its height and dividing by three. This means that we have the following formula:

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

We know that the base of a square pyramid is a square. Furthermore, we also know that the area of a square is found by squaring one of its side lengths. Therefore, we have the following formula:

where, l is the length of one side of the square base, and h is the height of the pyramid.

## Volume of a square pyramid – Examples with answers

The following examples can be used to practice applying the formula for the volume of square pyramids. Try to solve the examples yourself before looking at the answer.

### EXAMPLE 1

What is the volume of a square pyramid that has a height of 5 m and sides of length 4 m?

We have the following lengths

• Sides of the square, $latex l=4$
• Height of pyramid, $latex h=5$

Therefore, we use these values in the volume formula:

$latex V=\frac{1}{3}{{l}^2}\times h$

$latex V=\frac{1}{3}{{(4)}^2}\times (5)$

$latex V=\frac{1}{3}(16)\times (5)$

$latex V=26.67$

The volume is equal to 26.67 m³.

### EXAMPLE 2

If we have a pyramid with a height of 6 m and a square base with sides of 5 m, what is its volume?

From the question, we get the following information:

• Sides of the square, $latex l=5$
• Height of pyramid, $latex h=6$

By substituting these values into the volume formula, we have:

$latex V=\frac{1}{3}{{l}^2}\times h$

$latex V=\frac{1}{3}{{(5)}^2}\times (6)$

$latex V=\frac{1}{3}(25)\times (6)$

$latex V=50$

The volume is equal to 50 m³.

### EXAMPLE 3

What is the volume of a pyramid that has a height of 9 m and a square base with sides of length 8 m?

We observe the following information:

• Sides of the square, $latex l=8$
• Height of pyramid, $latex h=9$

Using these values in the volume formula, we have:

$latex V=\frac{1}{3}{{l}^2}\times h$

$latex V=\frac{1}{3}{{(8)}^2}\times (9)$

$latex V=\frac{1}{3}(64)\times (9)$

$latex V=192$

The volume is equal to 192 m³.

### EXAMPLE 4

A square pyramid has a volume of 96 m³. If its sides are 6 m long, what is the length of its height?

We have the following values:

• Sides of the square, $latex l=6$
• Volume, $latex V=96$

In this case, we have the volume and we want to find the length of the height of the pyramid. Therefore, we use the volume formula and solve for h:

$latex V=\frac{1}{3}{{l}^2}\times h$

$latex 96=\frac{1}{3}{{(6)}^2}\times h$

$latex 96=\frac{1}{3}(36)\times h$

$latex 96=12 h$

$latex h=8$

The length of the height is equal to 8 m.

### EXAMPLE 5

If a square pyramid has a volume of 75 m³ and a height of 9 m, what is the length of its height?

We have the following information:

• Height, $latex h=9$
• Volume, $latex V=75$

We use the volume formula and solve for it:

$latex V=\frac{1}{3}{{l}^2}\times h$

$latex 75=\frac{1}{3}{{l}^2}\times (9)$

$latex 75=3{{l}^2}$

$latex {{l}^2}=25$

$latex l=5$

The length of the sides is equal to 5 m.

## Volume of a square pyramid – Practice problems

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

#### If a square pyramid has a volume of 100 $latex {{m}^3}$ and sides of length of 5m, what is the length of its height?  