Exterior Angles of a Polygon

Walk once around the edge of any convex polygon, and at each corner you make a small turn. That turn is the exterior angle at that corner — the angle between one side and the extension of the side before it. By the time you arrive back where you started, you have turned through one full circle. So however many sides the polygon has, the exterior angles always add up to the same thing:

\text{sum of exterior angles} = 360^\circ

This builds straight on the angle sum of a polygon: each interior angle and its exterior angle together make a straight line.

Walk once around a convex polygon, turning at each corner. Then:

the turns are one full revolution, so the exterior angles always sum to 360^\circ — whatever n is;
at every corner the interior angle and its exterior angle lie on a straight line, so \text{interior} + \text{exterior} = 180^\circ;
for a regular n-gon all the turns are equal, so each exterior angle is \dfrac{360^\circ}{n}.

Why it works

No formula needed — just take a walk. Extend each side a little and watch the turn you make at every corner. Step through it.

The five turns return you to your starting direction, so together they make a complete 360^\circ. The same walk works for a triangle, a hexagon, or any convex polygon — the total is always 360^\circ.