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:
