**Sum of Interior Angles of a Polygon :**

In this section, we are going to learn, how to find the sum of interior angles of a polygon.

**Formula to find the sum of interior angles of a n-sided polygon (when number of sides is given) : **

**(n - 2) ⋅ 180°**

(The above formula can be applied to both regular and irregular polygons)

**Formula to find the sum of interior angles of a n-sided ****regular**** polygon (when number of sides "n" and measure of each interior**** are given) : **

**n ⋅ measure of each interior angle**

**Formula to find the measure of each interior angle ****of a n-sided ****regular**** polygon (****when the sum of interior angles and number of sides "n" ****are given) : **

**Sum of interior angles / n**

**Formula to find the number of sides of a ****regular**** polygon (when the sum of interior angles and the measure of each angle are ****given) : **

**aSum of interior angles / Measure of each interior aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaangle**

**Sum of exterior angles of a polygon is :**** **

**360**°

**Formula to find the number of sides of a**** ****regular**** polygon (****when the measure of each exterior angle is known)**** : **

**360 / Measure of each exterior angle**

**In any polygon, the sum of an interior angle and its corresponding exterior angle is : **

**180°**

**Note : **

If a polygon has "n" number of sides, then it would have "n" number of vertices, "n" number of interior angles and "n" number of exterior angles.

**Regular Polygon : **

**A regular polygon has sides of equal length, and all its interior and exterior angles are of same measure. **

**Irregular Polygon :**

**An irregular polygon can have sides of any length and angles of any measure. **

**Interior Angle :**

An interior angle of a polygon is an angle inside the polygon at one of its vertices.

**Exterior Angle :**

An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side.

**Problem 1 :**

Find the value of "x" in the diagram given below.

**Solution : **

The above diagram is an irregular polygon of 5 sides.

Formula to find the sum of interior angles of a n-sided polygon is

= (n - 2) ⋅ 180°

By using the formula, sum of the interior angles of the above polygon is

= (5 - 2) ⋅ 180°

= 3 ⋅ 180°

= 540° ------(1)

By using the angles, sum of the interior angles of the above polygon is

= 58° + 100° + 112° + 25° + x°

= 295° + x° ------(2)

From (1) and (2), we get

295° + x° = 540°

295 + x = 540

Subtract 295 from both sides.

x = 245

Hence, the value of "x" is 245.

Let us look at the next problem on "Sum of interior angles of a polygon"

**Problem 2 :**

Find the value of "x" in the diagram given below.

**Solution : **

The above diagram is an irregular polygon of 6 sides (Hexagon) with one of the interior angles as right angle.

Formula to find the sum of interior angles of a n-sided polygon is

= (n - 2) ⋅ 180°

By using the formula, sum of the interior angles of the above polygon is

= (6 - 2) ⋅ 180°

= 4 ⋅ 180°

= 720° ------(1)

By using the angles, sum of the interior angles of the above polygon is

= 120° + 90° + 110° + 130° + 160 + x°

= 610° + x° ------(2)

From (1) and (2), we get

610° + x° = 720°

610 + x = 720

Subtract 610 from both sides.

x = 110

Hence, the value of "x" is 110.

Let us look at the next problem on "Sum of interior angles of a polygon"

**Problem 3 :**

Find the measure of each interior angle of the regular polygon given below.

**Solution : **

Let us count the number of sides of the polygon given above.

So, the above regular polygon has 9 sides.

Formula to find the sum of interior angles of a n-sided polygon is

= (n - 2) ⋅ 180°

By using the formula, sum of the interior angles of the above polygon is

= (9 - 2) ⋅ 180°

= 7 ⋅ 180°

= 1260°

**Formula to find the measure of each interior angle ****of a n-sided regular ****polygon is **** **

= Sum of interior angles / n

Then, we have

= 1260° / 9

= 140°

Hence, the measure of each interior angle of the given regular polygon is 140°.

Let us look at the next problem on "Sum of interior angles of a polygon"

**Problem 4 :**

What is the measure of each interior angle of a regular decagon ?

**Solution : **

Decagon is a 10-sided polygon.

Formula to find the sum of interior angles of a n-sided polygon is

= (n - 2) ⋅ 180°

By using the formula, sum of the interior angles of the given decagon (10-sided polygon) is

= (8 - 2) ⋅ 180°

= 8 ⋅ 180°

= 1440°

**Formula to find the measure of each interior angle ****of a n-sided regular ****polygon is **** **

= Sum of interior angles / n

Then, we have

= 1440° / 10

= 144°

Hence, the measure of each interior angle of the given regular decagon is 144°.

Let us look at the next problem on "Sum of interior angles of a polygon"

**Problem 5 :**

Each exterior angle of a regular polygon measures 30°. How many sides does the polygon have ?

**Solution : **

Formula to find the number of sides of a regular polygon (when the measure of each exterior angle is known) :

= 360 / Measure of each exterior angle

Then, we have

= 360 / 30

= 12

Hence, the given polygon has 12 sides.

**Problem 6 :**

Each interior angle of a regular polygon measures 160°. How many sides does the polygon have ?

**Solution : **

In any polygon, the sum of an interior angle and its corresponding exterior angle is 180°.

That is,

Interior angle + Exterior Angle = 180°

160° + Exterior Angle = 180°

Exterior angle = 20°

So, the measure of each exterior angle is 20°.

Formula to find the number of sides of a regular polygon (when the measure of each exterior angle is known) :

= 360 / Measure of each exterior angle

Then, we have

= 360 / 20

= 18

Hence, the given polygon has 18 sides.

After having gone through the stuff given above, we hope that the students would have understood, how to find the sum of interior angles of a polygon

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