The area of a pentagon is the region that is bounded by the sides of the pentagon, that is, it is the region occupied by the figure in the two-dimensional plane. The formula for the area of a pentagon is found by dividing the area of the pentagon into five isosceles triangles and calculating accordingly.

There are two main formulas that we can use to calculate its area, one takes into account the length of the sides and the apothem and the other only takes into account the length of the sides.

## Formula to find the area of a pentagon

The formula for the area of the pentagon is derived by dividing the pentagon into five isosceles triangles as in the following image:

We know that the area of a triangle is $latex A=\frac{1}{2}bh$, where *b* is the length of the triangle’s base and *h* is the length of the triangle’s height. In the isosceles triangles above, the length of the base is equal to the length of one of the sides of the pentagon, which we can represent with *s*.

The height of the triangle is equal to the apothem of the pentagon, which we can represent with *a*. Therefore, the height of each triangle in the pentagon is $latex A=\frac{1}{2} sa$. Since we have five triangles, the area of the pentagon is:

$latex A= \frac{5}{2} \times s \times a$ |

Alternatively, the area of the pentagon can be found with the following formula:

$latex A=\frac{1}{4}\sqrt{5(5+2\sqrt{5}){{s}^2}}$ |

where *s* is the length of one of the sides of the pentagon. This formula is a bit more complicated, but it allows us to find the area of a pentagon simply by knowing the length of one of its sides.

## Area of a pentagon – Examples with answers

The following examples are solved using the formulas detailed above. Each exercise has its respective solution, where you can look at the process and reasoning used.

**EXAMPLE 1**

What is the area of a pentagon that has sides of length 10 m and an apothem of length 6.88 m?

##### Solution

The question gives us the following information:

- Sides, $latex s=10$ m
- Apothem, $latex a = 6.88$ m

Using these values in the area formula, we have:

$latex A= \frac{5}{2}sa$

$latex A= \frac{5}{2}(10)(6.88)$

$latex A=172$

The area of the pentagon is 172 m².

**EXAMPLE 2**

A pentagon has sides of length 8 m and an apothem of length 5.51 m. What is its area?

##### Solution

We have the following values:

- Sides, $latex s=8$ m
- Apothem, $latex a=5.51$ m

We can substitute these values in the formula:

$latex A= \frac{5}{2}sa$

$latex A= \frac{5}{2}(8)(5.51)$

$latex A=110.2$

The area of the pentagon is 110.2 m².

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**EXAMPLE 3**

The area of a pentagon is 84.3 m² and the length of its sides is 7 m. What is the length of its apothem?

##### Solution

In this case, we start with the area of the pentagon and we want to find the apothem. The question gives us the following information:

- Area, $latex A=84.3$ m²
- Sides, $latex s = 7$ m

We substitute these values and solve for *a*:

$latex A= \frac{5}{2}sa$

$latex 84.3= \frac{5}{2}(7)a$

$latex 168.6=(5)(7)a$

$latex 168.6=35a$

$latex a=4.82$

The length of the apothem is 4.82 m.

**EXAMPLE 4**

What is the area of a pentagon that has sides of length 5 m?

##### Solution

In this case, we only have the length of the pentagon’s sides, so we have to use the second area formula:

$latex A= \frac{1}{4}\sqrt{5(5+2\sqrt{5}){{s}^2}}$

$latex A= \frac{1}{4}\sqrt{5(5+2\sqrt{5}){{(5)}^2}}$

$latex A=43.01$

The area of the pentagon is 43.01 m².

**EXAMPLE 5**

What is the area of a pentagon that has sides of length 20 m?

##### Solution

We use the second formula of the area with the value $latex s = 20$:

$latex A= \frac{1}{4}\sqrt{5(5+2\sqrt{5}){{s}^2}}$

$latex A= \frac{1}{4}\sqrt{5(5+2\sqrt{5}){{(20)}^2}}$

$latex A=688.19$

The area of the pentagon is 688.19 m².

## Area of a pentagon – Practice problems

Use the two formulas for the area of pentagons seen above to solve the following problems. Select an answer and check it to make sure you selected the correct one.

## See also

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

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