Sum of angles in an equilateral $n$-polygon if $n$ approaches infinity

Question

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)?${}$

With the inside angles I mean the angeles yhat always sum up to 360 degrees. In a octagon the sum of the inside angles is 8x45 =360 degrees, but the sum of the ouside angles (inside the polygon) is 8x135=1080 degrees. Of any polygon the sum of the inside angles is 360 degrees. The more an polygon is going to look like a circle (with again the inner degrees of 360 degrees, the more the sum of the outer (but, again, inside the polygon) degrees reaches for infinity. Are in the case of an infinite polygon (a circle) the 360 degrees and the inifinite degrees some kind of complement of eachother? — Deschele Schilder, Feb 06 '16 at 16:19

score 0 · Answer 1 · answered Jan 26 '16 at 23:14

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.

Sum of angles in an equilateral $n$-polygon if $n$ approaches infinity

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