Evelina,

Here is one way to solve the problem. I will include some pictures - because my reasoning is very much visual

In general, for any (convex) polygon, the sum of the exterior angles is 360 degrees
Here is why this is true: Consider walking around the polygon. You turn through the exterior angle at each vertex.

When you go all the way around, then you are back facing the same way you started - and have done one complete 'turn'.

Now, if the polygon is regular, all the individual corner turns are the same size. With K turns, each turn would would (360/K).

There are alternative ways (with many more steps) which convert through using interior angles - but they have no advantage that I can see!

Walter Whiteley
York University