# How to Find the Area of a Pentagon (Formula & Example)

## Area of a pentagon

The area of a pentagon is the space inside its five straight sides. Most of the time, you will be tasked with finding the area of a regular pentagon, so this lesson will not cover irregular pentagons.

A regular pentagon has equal sides and congruent angles. There are a couple of methods you can use to calculate the area of a regular pentagon. One method uses a side length and length of the apothem.

### Apothem of a pentagon

The apothem of a pentagon is a line segment from the center of the pentagon to a side of the pentagon. The apothem is perpendicular to the side. All regular polygons have an apothem. For a polygon of * n* sides, there are

*apothems.*

**n**## Area of a pentagon formula

To find the area of a pentagon with the apothem, a, and one side length, s, you use the area of a pentagon formula:

What if you do not know the apothem of your pentagon? You can still find the area of a regular pentagon if you know:

A little trigonometry

The length of one side

Each interior angle measures

**108°**

You know that each interior angle measures **108°** because you know a few things about exterior angles and polygons. You know that:

The sum of the exterior angles of any polygon add up to

**360°**The exterior angle is the supplement of the interior angle (

**interior + exterior = 180°**)

To find the measure of each exterior of a regular polygon, you divide **360°** by the number of sides. For a pentagon that is $\frac{360°}{5}$. This tells us each exterior angle is **72°.**

Now we can use that to determine the measure of each interior angle. Remember, the exterior angle and interior angle must add to **180°**, so we have **180° − 72° = 108°**. Each interior angle equals **108°**.

### How to find the apothem and area of a pentagon

Using the length of one side and the measure of the interior angle, let's calculate the apothem length and find the area of a regular pentagon.

Let's say we have a pentagon with a side length of **4 cm**. Divide the pentagon into five isosceles triangles, each with a base formed by the pentagon's sides. Now, divide any one of those triangles into two right triangles:
You now know all this about the right triangle:

The length of the triangle’s short leg ($\frac{1}{2}$ the pentagon’s side)

The right angle (

**90°**angle) is opposite the hypotenuse (perpendicular bisector of the side)**36°**acute angle opposite the short leg**360°**divided among**10**right triangles)**54°**acute angle opposite the long leg ($\frac{1}{2}$ of the**108°**interior angle)

The tangent of an angle (here, our **36°** angle) is the opposite side (the short leg) divided by adjacent side (the long leg, which is both the height of the triangle * and* the apothem of the pentagon):

The **tan(36°)** is approximately **0.727**, so we have the opposite side (the short leg) of **2 cm** divided by **0.727**:

With the height, **h**, of the triangle now established and knowing the triangle’s base ($\frac{1}{2}$; the pentagon’s side), * b*, you can now apply the formula for the area of a triangle:

We have **10** such right triangles, so we modify the triangle area formula and calculate the area of our regular pentagon:

The one half and the ten can be combined:

Now, we plug in the numbers that we know for the base and height:

And we arrive at our answer:

The total area of the pentagon is $27.5{cm}^{2}$. Area is always expressed in units squared or square units.