Sum of Angles Formula

We use the sum of angles formula to determine the sum of interior angles of a polygon. The sum of angles in a polygon depends on the number of vertices it has. When there is a polygon with four or more than four sides, we draw all the possible diagonals from one vertex. Then the polygon is broken into several non-overlapping triangles.  Let us learn about the sum of angles formula with a few examples in the end.

What Is the Sum of Angles Formula?

The formula for interior angles can be determined by multiplying the number of triangles by 180°and the total number of triangles is two less than the number of sides of a polygon, always. The sum of angles formula of a given polygon can be expressed as

• The sum of the interior angles of a given polygon = (n − 2) × 180°, where n = the number of sides of the polygon.
• The sum of exterior angles of a given polygon = 360°

(n-2) denotes the number of triangles that a polygon could be divided into and 180° is for counting the number of degrees in a triangle.

Let us have a look at a few solved examples on the sum of angles in a polygon to understand the concept better.

Example on Sum of Angles Formula

Example 1: George cuts a piece of paper into a regular pentagonal polygon and he wants to know the sum of interior angles of the regular pentagon. Find the sum of interior angles of a regular pentagon for George.

Solution:

To find: The sum of interior angles of a regular pentagon.
Sides of pentagon (n)  = 5 (given)
Using sum of angles formula,
The sum of its interior angle of a given polygon (S) = ( n − 2) × 180°
S = ( 5 − 2) × 180°
= (3) × 180°
= 540°

Answer: The sum of the interior angles of a regular pentagon is 540°

Example 2: Using the angle sum formula verify that sum of interior angles of a triangle is 180 degrees.

Solution: To verify the sum of interior angles of a triangle is 180 degrees.

Angle sum formula = ( n − 2) × 180°.
Sides of a triangle = 3
Putting the value of n = 2 in angle sum formula we have, (3 − 2) × 180° = 180°

Answer: Hence proved the sum of interior angles of a triangle is 180 degrees.

Example 3: Harry wants to find the interior angle of the hexagon brick. Help him to find the measure of each interior angle of a regular hexagon. 

Solution: To find: The measure of each interior angle of a regular hexagon
Sides of Hexagon (n)  = 6 (given)
Using the sum of angles formula,
The sum of the interior angles of a given polygon (S) = ( n − 2) × 180°
S = ( 6 − 2) × 180°
= (4) × 180°
S = 720°
The measure of each interior angle of a regular hexagon = 720°/6 = 120°

Answer: The measure of each interior angle of a regular hexagon brick is 120°