# Sum of Interior angles of Polygon

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

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