The volume of a pentagonal prism is equal to the space occupied by the prism in all three dimensions. This volume can be calculated by multiplying the area of the pentagonal base by the height of the prism. In turn, the area of a pentagon can be found using the lengths of the apothem and one of its sides. It is also possible to find the volume of these prisms using only the length of the sides and the height of the prism, but this requires a slightly more complicated formula.

Here, we will learn about the formulas that we can use to calculate the volume of pentagonal prisms. In addition, we will use these formulas to solve some practice problems.

## Formula to find the volume of a pentagonal prism

We can find the volume of a pentagonal prism by multiplying the area of the base by the height of the prism. Recall that we can use the apothem to calculate the area of polygons easily. Therefore, we have the following formula:

$latex V = \frac{5}{2} alh$

where *a* represents the length of the apothem, *l* represents the length of the sides of the pentagonal base and *h* represents the length of the height of the prism.

Furthermore, we can also find the volume of a prism using only the length of its height and the length of one of the sides of its hexagonal base. For this, we use the following formula:

$latex V=\frac{1}{4}\sqrt{5(5+2\sqrt{5}})~{{l}^2}h$

This formula is more complicated, but we can approximate it to the following expression:

$latex V=1.72{{l}^2}h$

## Volume of a pentagonal prism – Examples with answers

The following examples put into practice the use of the pentagonal prism volume formulas seen above. Each of the examples has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

What is the volume of a prism that has a height of 6 m and a base with sides of length 8 m and an apothem of 5.5 m?

##### Solution

We have the following information:

- Prism height, $latex h=6$
- Pentagon sides, $latex l=8$
- Apothem, $latex a=5.5$

Using the first volume formula with this information, we have:

$latex V=\frac{5}{2}alh$

$latex V=\frac{5}{2}(5.5)(8)(6)$

$latex V=660$

The volume of the prism is 660 m³.

**EXAMPLE 2**

A prism has a height of 8 m and its base is a pentagon with sides of length 9 m and an apothem of 6.2 m. What is its volume?

##### Solution

We recognize the following values:

- Prism height, $latex h=8$
- Pentagon sides, $latex l=9$
- Apothem, $latex a=6.2$

We substitute these values in the first formula of the volume:

$latex V=\frac{5}{2}alh$

$latex V=\frac{5}{2}(6.2)(9)(8)$

$latex V=1116$

The volume of the prism is 1116 m³.

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**EXAMPLE 3**

If a prism has a height of 10 m and a pentagonal base with sides of length 5 m, what is its volume?

##### Solution

We have the following values:

- Prism height, $latex h=10$
- Pentagon sides, $latex l=5$

We have to use the second volume formula with these values:

$latex V=1.72{{l}^2}h$

$latex V=1.72{{(5)}^2}(10)$

$latex V=430$

The volume of the prism is 430 m³.

**EXAMPLE 4**

A prism has a pentagonal base with sides of length 3 m. If its height is 5 m, what is its volume?

##### Solution

We have the following:

- Prism height, $latex h=5$
- Pentagon sides, $latex l=3$

Using the second formula with these values, we have:

$latex V=1.72{{l}^2}h$

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

$latex V=77.4$

The volume of the prism is 77.4 m³.

**EXAMPLE 5**

What is the volume of a prism that has a height of 12 m and a pentagonal base with sides of 6 m?

##### Solution

We have the following values:

- Prism height, $latex h=12$
- Pentagon sides, $latex l=6$

If we use the second formula, we have:

$latex V=1.72{{l}^2}h$

$latex V=1.72{{(6)}^2}(12)$

$latex V=743.04$

The volume of the prism is 743.04 m³.

## Volume of a pentagonal prism – Practice problems

Put into practice the use of the formulas for the volume of pentagonal prisms and solve the following problems. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about pentagonal prisms? Take a look at these pages:

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