It's obvious that in the case of an equilateral polygon, the number of angles between two sides increases in number, as are the angles themselves. Now the angle between two lines from both sides of one of the $n$ sides is clearly $\frac{360^\circ}{n}$.
But what about the angles at the outside of the $n$-polygon. Does their sum go to infinity, if $n$ goes to infinity (and the polygon becomes a circle)?${}$
The inner-angle sum becomes infinity, the outer-angle sum remains 360 degrees. This is trivial to verify, because the tangent vector representing outer-angle orientation traces through a full revolution, while each inner-angle approaches 180 degrees.