 # Interior Angles of a Polygon

An interior angle of a polygon is an angle formed inside the two adjacent sides of a polygon. Or, we can say that the angle measures at the interior part of a polygon are called the interior angle of a polygon.

## Interior Angles of a Polygon Formula

The interior angles of a polygon always lie inside the polygon. The formula can be obtained in three ways. Let us discuss the three different formulas in detail.

Method 1:

If “n” is the number of sides of a polygon, then the formula is given below:

Interior angles of a Regular Polygon = [180°(n) – 360°] / n

Method 2:

If the exterior angle of a polygon is given, then the formula to find the interior angle is

Interior Angle of a polygon = 180° – Exterior angle of a polygon

Method 3:

If we know the sum of all the interior angles of a regular polygon, we can obtain the interior angle by dividing the sum by the number of sides.

Interior Angle = Sum of the interior angles of a polygon / n

Where

“n” is the number of polygon sides

## Polygons Interior Angles Theorem

Below is the proof for the polygon interior angle sum theorem

Statement:

In a polygon of ‘n’ sides, the sum of the interior angles is equal to (2n – 4) × 90°.

To prove:

The sum of the interior angles = (2n – 4) right angles

Proof: ABCDE is a “n” sided polygon. Take any point O inside the polygon. Join OA, OB, OC.

For “n” sided polygon, the polygon forms “n” triangles.

We know that the sum of the angles of a triangle is equal to 180 degrees

Therefore, the sum of the angles of n triangles = n × 180°

From the above statement, we can say that

Sum of interior angles + Sum of the angles at O = 2n × 90° ——(1)

But, the sum of the angles at O = 360°

Substitute the above value in (1), we get

Sum of interior angles + 360°= 2n × 90°

So, the sum of the interior angles = (2n × 90°) – 360°

Take 90 as common, then it becomes

The sum of the interior angles = (2n – 4) × 90°

Therefore, the sum of “n” interior angles is (2n – 4) × 90°

So, each interior angle of a regular polygon is [(2n – 4) × 90°] / n

Note: In a regular polygon, all the interior angles are of the same measure.

## Interior Angles for Different Shapes

The interior angles of different polygons do not add up to the same number of degrees. Let us discuss the sum of interior angles for some polygons:

 Polygon Number of sides (n) Sum of interior angles (n – 2) × 180 Triangle 3 180° Quadrilateral 4 360° Pentagon 5 540° Hexagon 6 720°

### Interior Angles of a Polygon Example

Question: If each interior angle is equal to 144°, then how many sides does a regular polygon have?

Solution:

Given: Each interior angle = 144°

We know that,

Interior angle + Exterior angle = 180°

Exterior angle = 180°-144°

Therefore, the exterior angle is 36°

The formula to find the number of sides of a regular polygon is as follows:

Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior angle

Therefore, the number of sides = 360° / 36° = 10 sides

Hence, the polygon has 10 sides.

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