The surface area of a pentagonal prism is calculated by adding the areas of all the faces of the prism. In total, we have seven faces on a pentagonal prism: two pentagonal faces and five rectangular faces. Therefore, we can find an expression for the surface area by finding expressions for the area of the pentagonal and rectangular faces.

Here, we will learn about the formula that we can use to calculate the surface area of pentagonal prisms. In addition, we will practice using this formula with some problems.

## Formula to find the surface area of a pentagonal prism

The surface area of any three-dimensional figure is found by adding the areas of all its faces. In the case of pentagonal prisms, we have two pentagonal faces and five rectangular faces.

We can find the area of each pentagonal face using the formula $latex 3.44{{l}^2}$, where, *l* represents the length of one of the sides of the pentagonal face. Therefore, the area of both pentagonal faces is $latex 6.88{{l}^2}$.

On the other hand, the area of each rectangular face is found using the formula *lh*, where, *l* is the length of one of the sides of the pentagonal face and *h* is the length of the height of the prism. Therefore, the area of the five rectangular areas is 5*hl*.

By adding these two parts, we have the formula for the surface area of a pentagonal prism:

$latex A_{s} = 6.88 {{l}^2}+5hl$

## Surface area of a pentagonal prism – Examples with answers

The formula for the surface area of pentagonal prisms is used to solve the following examples. Each example has its respective solution, where the process and reasoning used are detailed.

**EXAMPLE 1**

What is the surface area of a pentagonal prism with a height of 4 m and sides of length 4 m?

##### Solution

We have $latex h=4$ and $latex l=4$. Therefore, we use the formula for surface area with these values:

$latex A_{s}=6.88{{l}^2}+5lh$

$latex A_{s}=6.88{{(4)}^2}+5(4)(4)$

$latex A_{s}=110.1+80$

$latex A_{s}=190.1$

The surface area is equal to 190.1 m².

**EXAMPLE 2**

If a prism has a height of 9 m and a pentagonal base with sides of 5 m, what is its surface area?

##### Solution

We use the lengths $latex h=9$ and $latex l=5$ in the formula for surface area:

$latex A_{s}=6.88{{l}^2}+5lh$

$latex A_{s}=6.88{{(5)}^2}+5(5)(9)$

$latex A_{s}=172+225$

$latex A_{s}=395$

The surface area is equal to 395 m².

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**EXAMPLE 3**

What is the surface area of a pentagonal prism that has sides of 6 m and a height of 11 m?

##### Solution

We have the lengths $latex h=11$ and $latex l=6$, so we use these values in the formula for surface area:

$latex A_{s}=6.88{{l}^2}+5lh$

$latex A_{s}=6.88{{(6)}^2}+5(6)(11)$

$latex A_{s}=247.7+330$

$latex A_{s}=577.7$

The surface area is equal to 577.7 m².

**EXAMPLE 4**

If a pentagonal prism has sides that are 8 m long and 12 m high, what is its surface area?

##### Solution

We replace the lengths $latex h=12$ and $latex l=8$ in the formula for the surface area:

$latex A_{s}=6.88{{l}^2}+5lh$

$latex A_{s}=6.88{{(8)}^2}+5(8)(12)$

$latex A_{s}=440.3+480$

$latex A_{s}=920.3$

The surface area is equal to 920.3 m².

## Surface area of a pentagonal prism – Practice problems

Use the formula for the surface area of a pentagonal prism to 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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