Sum of the Exterior Angles of a Polygon

We will learn how to find the sum of the exterior angles of a polygon having n sides.

We know that, exterior angle + interior adjacent angle = 180°

So, if the polygon has n sides, then

Sum of all exterior angles + Sum of all interior angles = n × 180°

So, sum of all exterior angles = n × 180° - Sum of all interior angles

Sum of all exterior angles = n × 180° - (n -2) × 180°

                                      = n × 180° - n × 180° + 2 × 180°

                                      = 180°n - 180°n + 360°

                                      = 360°

Therefore, we conclude that sum of all exterior angles of the polygon having n sides = 360°

Therefore, measure of each exterior angle of the regular polygon = 360°/n

Also, number of sides of the polygon = 360°/each exterior angle

Solved examples on sum of the exterior angles of a polygon:

1. Find the number of sides in a regular polygon when the measure of each exterior angle is 45°.

Solution:

If the polygon has n sides,

Then, we know that; n = 360°/measure of each exterior angle

                                = 360/45

                                = 8

Therefore, the regular polygon has 8 sides.

2. The exteriors angles of a pentagon are (m + 5)°, (2m + 3)°, (3m + 2)°, (4m + 1)° and (5m + 4)° respectively. Find the measure of each angle.

Hints: The sum of all exterior angles of a polygon is 360°.

Solution:

We know, the sum of all exterior angles of a pentagon is 360°

Therefore, (m + 5)° + (2m + 3)° + (3m + 2)° + (4m + 1)° + (5m + 4)° = 360°

⇒ m + 5 + 2m + 3 + 3m + 2 + 4m + 1 + 5m + 4 = 360°

⇒ 15m + 15 = 360°

⇒ 15m = 360° - 15°                          

⇒ 15m = 345°    

⇒ m = 345°/15°

⇒ m = 23°

Therefore, the first angle = m + 5°           

                                   = 23° + 5°

                                   = 28°

Second angle = 2m + 3°

                   = 2° × 23° + 3°

                   = 46° + 3°

                    = 49°

Third angle = 3m + 2

                = 3° × 23° + 2°

                = 69° + 2°

                 = 71°

Fourth angle = 4m + 1

                  = 4° × 23° + 1°

                  = 92° +1°

                  = 93°

Fifth angle = 5m + 4°

                = 5° × 23° + 4°

                = 115° + 4°

                = 119°

● Polygons

Polygon and its Classification

Terms Related to Polygons

Interior and Exterior of the Polygon

Convex and Concave Polygons

Regular and Irregular Polygon

Number of Triangles Contained in a Polygon

Angle Sum Property of a Polygon

Problems on Angle Sum Property of a Polygon

Sum of the Interior Angles of a Polygon

Sum of the Exterior Angles of a Polygon

7th Grade Math Problems 

8th Grade Math Practice 

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