Exterior angles are easy to define for convex polygons. They also lie outside the polygon, making it intuitive as to why they are called "exterior". But I'm a bit confused when we talk about exterior angles in concave polygons.

Let's say a 7 sided polygon has only one reflex interior angle (thus, making it concave). How is the exterior angle at the vertex containing the reflex angle defined? Because if one extends one of the sides of the polygon at that vertex, it enters the polygon. So the angle is no longer external? So not exterior? I'm really confused here. Any help?

  • $\begingroup$ In my opinion, an exterior angle of a polygon it's a futile concept, which bothers to think right. There is a concept as an angle of the polygon $A_1A_2...A_n$, which is $\angle A_1A_2A_3,$ for example. There is a concept as an exterior angle of a triangle, but it's an unique situation. $\endgroup$ – Michael Rozenberg Jul 22 at 4:54

Imagine each vertex the center of a small disk. The circle bounding the disk represents $360^\circ$. One arc of the circle is interior to the polygon, one arc exterior. The boundary of the polygon splits the disk into inside and outside. The total interior arc length is the internal angle; the total exterior arc length is the exterior angle:

          Internal + external $= 360^\circ$  :  $90^\circ+270^\circ$ and $60^\circ+300^\circ$.

  • $\begingroup$ could you please explain the "internal + external" line a bit more? I don't seem to get how that clears my confusion :) $\endgroup$ – Apekshik Panigrahi Jul 21 at 4:06

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