The volume of a pentagonal pyramid represents the total space occupied by the pyramid in three-dimensional space. This is a three-dimensional measurement, so we use cubic units to represent it. The volume is calculated by multiplying the area of the base by the length of the height of the pyramid. This means that we need an expression for the area of a pentagon before calculating the volume.

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

## Formula to find the volume of a pentagonal prism

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

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

In turn, these pyramids have a pentagonal base and the area of a pentagon is calculated using the following formula:

$latex A=1.72{{l}^2}$

where *l* is the length of one of the sides of the pentagon.

This formula is derived by dividing the pentagon into five triangles and finding the area of each triangle separately.

Using the expression for the area of a given pentagon, the formula for the volume of a pyramid becomes:

$latex V=\frac{1.72}{3}{{l}^2}h$ |

where *l* is the length of one of the sides of the pentagonal base and *h* is the length of the height of the pyramid.

## Volume of a pentagonal pyramid – Examples with answers

Each of the following examples is solved using the formula for the volume of pentagonal pyramids. It is recommended that you try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the volume of a pyramid that has a pentagonal base of 1 m and a height of 3 m?

##### Solution

We have the following:

- Sides of the pentagon, $latex l=1$
- Height, $latex h=3$

We use the volume formula with these values:

$latex V=\frac{1.72}{3}{{l}^2}h$

$latex V=\frac{1.72}{3}{{(1)}^2}(3)$

$latex V=\frac{1.72}{3}(1)(3)$

$latex V=1.72$

The volume is 1.72 m³.

**EXAMPLE 2**

A pentagonal pyramid has a base with sides of a length of 2 m and a height of 5 m. What is its volume?

##### Solution

We have these values:

- Sides of the pentagon, $latex l=2$
- Height, $latex h=5$

We substitute these values in the volume formula:

$latex V=\frac{1.72}{3}{{l}^2}h$

$latex V=\frac{1.72}{3}{{(2)}^2}(5)$

$latex V=\frac{1.72}{3}(4)(5)$

$latex V=11.47$

The volume is 11.47 m³.

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**EXAMPLE 3**

If a pentagonal pyramid has sides that are 3 m long and 6 m high, what is its volume?

##### Solution

We have the following:

- Sides of the pentagon, $latex l=3$
- Height, $latex h=6$

We use the volume formula with these values:

$latex V=\frac{1.72}{3}{{l}^2}h$

$latex V=\frac{1.72}{3}{{(3)}^2}(6)$

$latex V=\frac{1.72}{3}(9)(6)$

$latex V=30.96$

The volume is 30.96 m³.

**EXAMPLE 4**

A pyramid has a pentagonal base with sides of length 5 m and a height of 12 m. What is its volume?

##### Solution

From the question, we get the following lengths:

- Sides of the pentagon, $latex l=5$
- Height, $latex h=12$

If we substitute these values in the volume formula, we have:

$latex V=\frac{1.72}{3}{{l}^2}h$

$latex V=\frac{1.72}{3}{{(5)}^2}(12)$

$latex V=\frac{1.72}{3}(25)(12)$

$latex V=172$

The volume is 172 m³.

## Volume of a pentagonal pyramid – Practice problems

Use the pentagonal pyramid volume formula to solve the following practice problems. If you need help with these, you can look at the solved examples above.

## See also

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

- Volume of a Square Pyramid – Formulas and Examples – Mechamath
- Surface Area of a Square Pyramid – Formulas and Examples – Mechamath
- Volume of a Hexagonal Pyramid – Formulas and Examples – Mechamath
- Surface Area of a Hexagonal Pyramid – Formulas and Examples – Mechamath
- Surface Area of a Pentagonal Pyramid – Formulas and Examples – Mechamath
- Parts of a Geometric Pyramid – Mechamath

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