In this chapter we will learn about the formula for calculating sum of interior angle of polygon with solved example.

Let us first review the basics of polygon and interior angles.

## What are polygons ?

Polygons are** two dimensional closed shape formed by straight lines**.

There are infinite types of polygons in geometry.

Polygons are named on the basis of number of sides.

**For example**;

3 side polygon is known as Triangle.

8 side polygon is known as Octagon.

## What are interior angles in Polygon ?

The **angles that lie inside the polygon shape** are called interior angles.

For any polygon, there are as many interior angles as the number of sides.

For example, observe the above image.

See that a quadrilateral has 4 sides and also 4 interior angles.

Similarly, a hexagon has 6 sides and 6 interior angles.

## Sum of Interior Angles of a Polygon

You can easily find the sum of all interior angles of any polygon using the following formula;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}

You have to memorize the above formula as questions are directly asked from the above expression.

Let us see some solved examples for the above formula.

## Sum of Interior Angles in Polygon – Solved Examples

(01) Find the sum of all interior angles of below polygon.

**Solution**

The given polygon has 8 sides and angle.

Using the angle sum formula;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 8\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 6\ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1080\ degree}

Hence, the sum of all interior angle is **1080 degree**.

**(02) Find the sum of interior angle of polygon with 31 sides**.

**Solution**

Since the polygon has 31 side, the value of n = 31.

Using the angle sum formula of polygon;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 31\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 29\ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 5220\ degree}

Hence, the sum of interior angles with 31 sides is** 5220 degree.**

**(03) Find the sum of interior angles of undecagon ?**

**Solution**

Undecagon is a polygon with 11 sides.

Using sum of interior angle formula for polygon;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 11\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 9\ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1620\ degree}

Hence, the sum of interior angles of undecagon is **1620 degree.**

**(04) The sum of interior angle of a polygon is 2340 degree. Find the number of side of given polygon**.

**Solution**

Total sum of interior angle = 2340

Using the sum of interior angle formula;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{2340\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{\frac{2340}{180} \ =\ ( n\ -2) \ }\\\ \\ \mathtt{13\ =\ n\ -\ 2}\\\ \\ \mathtt{n\ =\ 13\ +\ 2}\\\ \\ \mathtt{n\ =\ 15}

Hence, the given polygon has **15 sides.**

**(05) The sum of interior angle of a regular polygon is 540 degree. Find the value of one interior angle.**

**Solution**

Sum of interior angle = 540 degree

Using sum of angle formula for polygon;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{540\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{\frac{540}{180} \ =\ ( n\ -2) \ }\\\ \\ \mathtt{3\ =\ n\ -\ 2}\\\ \\ \mathtt{n\ =\ 3\ +\ 2}\\\ \\ \mathtt{n\ =\ 5}

Hence, the polygon has total of 5 sides and angles.

It is given that the given shape is a regular polygon. We know that in regular polygon, the value of all interior angles are equal.

So divide the total angle measurement by 5 to get interior angle value;

⟹ 540 / 5

⟹ 108 degree.

Hence, the value of each interior angle is** 108 degree**