# Sum of the Interior Angles of a Polygon

We will learn how to find the sum of the interior angles of a polygon having n sides.

We know that if a polygon has ‘n’ sides, then it is divided into (n – 2) triangles.

We also know that, the sum of the angles of a triangle = 180°.

Therefore, the sum of the angles of (n – 2) triangles = 180 × (n – 2)

= 2 right angles × (n – 2)

= 2(n – 2) right angles

= (2n – 4) right angles

Therefore, the sum of interior angles of a polygon having n sides is (2n – 4) right angles.

Thus, each interior angle of the polygon = (2n – 4)/n right angles.

Now we will learn how to find the find the sum of interior angles of different polygons using the formula.

 Name Figure Number of Sides Sum of interior angles (2n - 4) right angles Triangle 3 (2n - 4) right angles= (2 × 3 - 4) × 90°= (6 - 4) × 90°= 2 × 90°= 180° Quadrilateral 4 (2n - 4) right angles= (2 × 4 - 4) × 90°= (8 - 4) × 90°= 4 × 90°= 360° Pentagon 5 (2n - 4) right angles= (2 × 5 - 4) × 90°= (10 - 4) × 90°= 6 × 90°= 540° Hexagon 6 (2n - 4) right angles= (2 × 6 - 4) × 90°= (12 - 4) × 90°= 8 × 90°= 720° Heptagon 7 (2n - 4) right angles= (2 × 7 - 4) × 90°= (14 - 4) × 90°= 10 × 90°= 900° Octagon 8 (2n - 4) right angles= (2 × 8 - 4) × 90°= (16 - 4) × 90°= 12 × 90°= 1080°


Solved examples on sum of the interior angles of a polygon:

1. Find the sum of the measure of interior angle of a polygon having 19 sides.

Solution:

We know that the sum of the interior angles of a polygon is (2n  - 4) right angles

Here, the number of sides = 19

Therefore, sum of the interior angles = (2 × 19 – 4) × 90°

= (38 – 4) 90°

= 34 × 90°

= 3060°

2. Each interior angle of a regular polygon is 135 degree then find the number of sides.

Solution:

Let the number of sides of a regular polygon = n

Then the measure of each of its interior angle = [(2n – 4) × 90°]/n

Given measure of each angle = 135°

Therefore, [(2n – 4) × 90]/n = 135

⇒      (2n – 4) × 90 = 135n

⇒         180n – 360 = 135n

⇒        180n - 135n = 360

⇒                    45n = 360

⇒                       n = 360/45

⇒                       n = 8

Therefore the number of sides of the regular polygon is 8.

Polygons

Polygon and its Classification

Terms Related to Polygons

Interior and Exterior of the Polygon

Convex and Concave Polygons

Regular and Irregular Polygon

Number of Triangles Contained in a Polygon

Angle Sum Property of a Polygon

Sum of the Exterior Angles of a Polygon