I have a question regarding the sum of the exterior angles of an n-sided polygon (n-gon). I understand that for a cyclic n-gon, the sum of its exterior angles is always 360 degrees. However, I'm not sure how to prove this for a general n-gon.

I came across a solution that involves drawing n triangles within the n-gon and using the fact that each triangle has an interior angle sum of 180 degrees to find the sum of all the interior angles in the n-gon. To do this, you can start by drawing an n-gon and connecting each vertex to all other vertices to form n triangles. However, I'm having trouble visualizing how to draw the n-gon.

The solution suggests starting with a circle and marking its center point, and then drawing lines from the center to the edges of the circle to form the n-gon. Can anyone provide a more detailed explanation or another approach to solving this problem?

Thank you in advance for your help!


1 Answer 1


Does it have to be though. Imagine a non convex closed polygon with a lot of local minima and maxima. You can have any number of sides and the sum of angles can be more than 360. But if we limit ourselves to convex polygons, (intuitive reasoning) an enclosing circle is always 360, since we always come back to the point we started from, thus its always 360.

  • $\begingroup$ Does it have to be what? $\endgroup$
    – cricket900
    Apr 1, 2023 at 17:52
  • $\begingroup$ Does it have to be 360. Not necessarily. A non convex polygon can have a sum that is not 360. $\endgroup$
    – xyz
    Apr 2, 2023 at 8:16

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