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.

Example 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°. 

Example 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. 

Example 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°. 

Example 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 70°. 

Example 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. 

Example 6 :

The measure of one of the base angle of an isosceles triangle is 42 degree. The measure of an exterior angle at the vertex of the triangle is 

a) 42    b) 84    c) 96     d) 138

Solution :

Since the given triangle is isosceles triangle and one angle measure is 42 degree then the other angle measure will also be 42 degree.

Let x be the that one missing interior angle measure.

42 + 42 + x = 180

84 + x = 180

x = 180 - 84

x = 96

interior angle + exterior angle = 180

96 + exterior angle = 180

exterior angle = 180 - 96

= 84

So, option b is correct.

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