# SUM OF EXTERIOR ANGLES OF A POLYGON

## About "Sum of Exterior Angles of a Polygon"

Sum of Exterior Angles of a Polygon :

In this section, we are going to see the sum of exterior angles of a polygon.

In any polygon (regular or irregular), the sum of exterior angle 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

Formula to find the measure of each exterior angle of a regular polygon (when the number of sides "n" given)  :

360° / n

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

180°

## Regular and Irregular Polygons

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 and Exterior Angles of a Polygon

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.

## Sum of Exterior Angles of a Polygon - Practice Problems

Problem 1 :

Find the measure of each exterior angle of a regular decagon.

Solution :

Decagon is a 10-sided polygon.

In any polygon, the sum of exterior angles is

=  360°

Formula to find the measure of each exterior angle of a regular n-sided polygon is :

360° / n

Then, we have

=  360° / 10

=  36°

Hence, the measure of each exterior angle of a regular decagon is 36°.

Problem 2 :

If the measure of each exterior angle of a regular pentagon is (2x + 4)°, find the value of x.

Solution :

Pentagon is a 5-sided polygon.

In any polygon, the sum of exterior angles is

=  360°

Formula to find the measure of each exterior angle of a regular n-sided polygon is :

360° / n

Then, we have

=  360° / 5

=  72°

The measure of each exterior angle is 72°.

Given : The measure of each exterior angle of a regular pentagon is (2x + 4)°.

So, we have

(2x + 4)°  =  72°

2x + 4  =  72

Subtract 4 from both sides.

2x  =  68

Divide both sides by

x  =  34

Hence, the value of "x" is 34.

Problem 3 :

Find the measure of each exterior 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.

In any polygon, the sum of exterior angles is

=  360°

Formula to find the measure of each exterior angle of a regular n-sided polygon is :

360° / n

Then, we have

=  360° / 9

=  40°

Hence, the measure of each exterior angle of a regular polygon is 40°.

Problem 4 :

Find the measure of exterior angle corresponding to the interior angle x° in the irregular polygon given below.

Solution :

To find the measure of exterior angle corresponding to x° in the above polygon, first we have to find the value of x.

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

So, the measure of interior angle represented by x is 110°.

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

That is,

Interior angle + Exterior Angle  =  180°

Then, we have

x° + Exterior Angle  =  180°

110° + Exterior angle  =  180°

Exterior angle  =  70°

So, the measure of each exterior angle corresponding to x° in the above polygon is 20°.

Problem 5 :

In a polygon, the measure of each interior angle is (5x+90)° and exterior angle is (3x-6)°. 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°

(5x + 90)° + (3x - 6)°  =  180°

5x + 90 + 3x - 6  =  180

8x + 84  =  180

8x  =  96

x  =  12

Finding the measure of exterior angle :

Exterior angle  =  (3x-6)°

Exterior angle  =  (3 ⋅ 12 - 6)°

Exterior angle  =  (36 - 6)°

Exterior angle  =  30°

Formula to find the number of sides of a regular polygon is

=  360 / Measure of each exterior angle

Then, we have

=  360 / 30

=  12

Hence, the given polygon has 12 sides.

After having gone through the stuff given abovewe hope that the students would have understood, "Sum of exterior angles of a polygon"

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