Interior Angles of a Polygon

In Mathematics, an angle is defined as the figure formed by joining the two rays at the common endpoint. Angles are generally measured using degrees or radians. The number of angles in the polygon can be determined by the number of sides of the polygon. For example, a square is a polygon which has four sides. Thus, the number of angles formed in a square is four. In this article, we are going to discuss what are the interior angles for different types of polygon, formulas, and interior angles for different shapes.

What is Meant by 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. We know that the polygon can be classified into two different types, namely:

  • Regular Polygon
  • Irregular Polygon

For a regular polygon, all the interior angles are of the same measure. But for irregular polygon, each interior angle may have different measurements.

Sum of Interior Angles of a Polygon

Depends on the number of sides, the sum of the interior angles of a polygon should be a constant value. No matter if the polygon is regular or irregular, convex or concave, it will give some constant measurement depends on the number of polygon sides.

For example, a square has four sides, thus the interior angles add up to 360°.

A pentagon has five sides, thus the interior angles add up to 540°, and so on.

Therefore, the sum of the interior angles of the polygon is given by the formula:

Sum of the Interior Angles of a Polygon = 180 (n-2) degrees

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:

Interior Angles of a Polygon

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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