the polygon exterior angle sum

Imagine walking once all the way around the edge of a polygon, like patrolling a fenced field, always facing forward. At each corner you turn through some angle to follow the next side. By the time you return to your starting point facing your original direction, you have turned through one complete revolution. The total of those corner-turns is the exterior angle sum.

An exterior angle at a vertex is the angle between one side extended and the next side — equivalently the supplement of the interior angle there, since the interior angle and the exterior angle together form a straight line (they add to 180 degrees). The striking fact: for any convex polygon, taking one exterior angle at each vertex, the exterior angles sum to exactly 360 degrees — no matter how many sides the polygon has. The 'walking' argument is the proof: a full circuit is one full turn, 360 degrees. You can also derive it from the interior sum: n interior angles plus n exterior angles total n·180 (n straight lines), so the exterior sum is n·180 - (n-2)·180 = 360 degrees.

Because the total is fixed at 360, a regular n-gon has each exterior angle equal to 360/n degrees — for instance 60 degrees for an equilateral triangle, 90 for a square, 72 for a regular pentagon. This is often the fastest route to a regular polygon's angles: find the exterior angle 360/n, then the interior angle is 180 minus it. The caveat: this clean 360 holds for convex polygons with one exterior angle counted per vertex; for concave or self-crossing polygons you must track turns with sign (left turns positive, right turns negative), and the signed total is 360 times the number of times the boundary winds around — the turning number — which need not be 1.