Pentagons have a sum of interior angles of 540°. Therefore, in the case of regular pentagons, each interior angle measures 108°. Irregular pentagons have angles of different measures, but their sum is always equal to 540°.

Here, we will learn more about the interior angles of a pentagon. We will learn about the formula for internal angles and we will solve some exercises.

## Interior angles in a regular pentagon

A regular pentagon is characterized by having all its sides of the same length, that is, its sides are congruent. Also, a regular pentagon has all its interior angles with the same measure. The sum of all the interior angles of any pentagon is always equal to 540°.

Since the interior angles of a regular pentagon are equal, we have to divide 540° by 5 to find the measure of each interior angle. Therefore, we have:

540°÷5 = 108°

Each interior angle in a regular pentagon is equal to 108°.

In the following diagram, we can see a regular pentagon that has all its sides the same length and all its angles the same measure. We see that by adding the five interior angles of 108°, we get a total of 540°.

## Formula to find the angles in a pentagon

The formula for the sum of the interior angles of any polygon is as follows:

° |

where *n* is the number of sides of a polygon. Therefore, for a pentagon we use . Using this, the formula becomes °. This shows that the sum of the interior angles of a pentagon is equal to 540°.

## How to find a missing angle in a pentagon?

In the case of irregular pentagons, we can find the measure of a missing angle by adding the measures of all known angles and subtracting the result from 540°.

For example, the following pentagon has the angles 100°, 120°, 80°, 110°.

Therefore, to find the missing angle, we start by adding the given angles:

100°+120°+80°+110° = 410°

Now, we subtract the result from 540°:

540°-410° = 130°

The missing angle measures 130°.

Now let’s look at an example where we have to find more than one missing angle. Here, angles that are different are marked differently. For example, angles that have a double line are equal.

Since the angles that have double lines are equal, we have *a*=120°.

Furthermore, we also know that angles *b* and *c* are equal since they both have triple lines. To find the measure of these angles, we start by adding the angles that we know so far:

90°+120°+120° = 330°

Now, we subtract this from 540° to find the missing angles:

540°-330° = 210°

Since the two missing angles are equal, we divide 210° by 2 to get the measure of each. Therefore, we have *b*=105° and *c*=105°.

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## Solved examples of interior angles of a pentagon

### EXAMPLE 1

Find the missing angles in the pentagon below.

**Solution:** In this case, we have the angles 90°, 100°, 130°, and 110°. Therefore, we start by adding them:

90°+100°+130°+110° = 430°

Now, we subtract this from 540° to find the missing angle:

540°-430° = 110°

The missing angle measures 110°.

### EXAMPLE 2

Find the missing angles in the pentagon below.

**Solution:** Angles that have triple lines have the same measure. Therefore, we have *c*=100°.

Angles *a* and *b* also have the same measure since they both have double lines. To find its measure, we have to start by adding the known angles:

80°+100°+100° = 280°

Now, we subtract this from 540° and we have:

540°-280° = 260°

Since the angles *a* and *b* are the same, we divide the result by 2 and we have *a*=130° and *b*=130°.

## See also

Interested in learning more about angles of a pentagon and other polygons? Take a look at these pages:

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