Sum of Interior Angles of a Pentagon

The sum of interior angles of a pentagon is equal to 540°. This is true regardless of whether the pentagon is regular or irregular. In the case of regular pentagons, we can determine the measure of each interior angle by dividing the total sum by 5. However, in the case of irregular pentagons, we need to know the measures of other angles to calculate the measure of some missing angle.

GEOMETRY

Relevant for

Learning about the sum of the interior angles of pentagons.

See angles

GEOMETRY

Relevant for

Learning about the sum of the interior angles of pentagons.

See angles

Sum of interior angles of a pentagon

The sum of all the interior angles of any pentagon is always equal to 540°. This applies regardless of whether the pentagon is regular or irregular. This sum is obtained by applying the polygon angle sum formula:

where,  n is the number of sides of the polygon. In the case of a pentagon, we have $latex n = 5$. Therefore, using the formula:

$latex (n-2)\times 180$°

$latex =(5-2)\times 180$°

$latex =(3)\times 180$°

$latex =540$°

In turn, this formula is obtained considering that we can divide any polygon into triangles as in the following diagram:

For any polygon, we can form a total of n-2 triangles. Also, we know that each triangle has a sum of interior angles of 180°, so $latex (n-2) \times 180$° corresponds to the sum of the interior angles of the polygon.

How to find the measures of all interior angles of pentagons?

If the pentagon is regular, we know that all its sides have the same measure. Therefore, we can determine the measures of each interior angle simply by dividing the total sum by 5. Therefore, we have:

540°÷5 = 108°

The measure of each internal angle in a regular pentagon is equal to 108°.

In the case of irregular pentagons, the measures of their interior angles are different from each other. Therefore, to find the measure of some missing angle, we need to know the measures of the other angles.

For example, if we know that a pentagon has the angles 110°, 100°, 120°, and 90°, we can find the measure of the fifth angle by adding the known angles and subtracting from 540°. Therefore, we have:

110°+100°+120°+90° = 420°

⇒  540°-420° = 120°

The measure of the missing angle is 120°.

Examples of interior angles of pentagons

EXAMPLE 1

Determine the measure of the missing angle in the pentagon below.

Solution: We know that the sum of all the interior angles of a pentagon is equal to 540°. Therefore, we have to add all the known angles and subtract from 540°:

80°+120°+100°+110° = 410°

⇒  540°-410° = 130°

The measure of the missing angle is 130°.

EXAMPLE 2

Determine the measures of the missing angles in the pentagon below.

Solution: The angles that are represented with the same color and have the same number of lines are equal. Therefore, angles a and b are equal, as are the angles 100° and c. Therefore, we have that c=100°. To find the missing angle measures, we add the known angle measures and subtract from 540°:

80°+100°+100° = 280°

⇒  540°-280° = 260°

Since the two angles are equal, we divide the result by 2 to get the measure of each. Therefore, we have a=130° and b=130°.