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
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