Lets find out sum of exterior angles of a polygon:
The sum of exterior angles of a polygon is always equal to 360° provided we take 1 exterior angles of a polygon at each vertex. From this the exterior angle of a regular polygon can be calculated by dividing 360 with number of sides of a polygon. For example, exterior angle of a regular hexagon is always equal to 360/6 = 60°. Exterior angle for a regular triangle is equal to 360/3 = 120°. Similarly exterior angle for a regular pentagon is equal to 360/5 = 72° and that for a regular octagon is 360/8 = 45°. This shows that exterior angle of regular polygon keeps on decreasing as the number of sides increases in a polygon. This formula can be applied to any convex polygon. For a concave polygon, some of the angles has to be taken negative to cope up with the formula.
Theorem for deriving exterior angles of a polygon formula:
The polygon exterior angle sum theorem helps to find out the generalized exterior angles of a polygon formula. This theorem states that the sum of exterior angles of a convex polygon is equal to the difference between sum of angles of linear pairs and sum of interior angles of a polygon. From this theorem, Sum of exterior angles S = 180n – 180(n – 2), where n is number of sides of a polygon.
i.e. S = 180n – 180n + 360 = 360°.
The figure given above shows a Pentagon and its exterior angles. From the theorem, we have
∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 360°. This theorems helps to calculate exterior angles sum for a n-sided polygon where n is a finite number.
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