actually, the same 180° value would have appeared with any smooth curve, not specifically with a circle. The reason for this is that if you draw a line between a fixed point A on a curve and another point M on the same curve then, as the point M tends to A along the curve, the line (AM) converges (in a sense that can be made rigorous) to the tangent line to the curve at A. This tangent line is the same if M tends to A from the left or the right, hence the flat angle (the angle between the two directions of the tangent at A).
In three dimensions, on a smooth surface, you have a similar property, namely that the line (AM) would get nearer and nearer to the tangent plane as M tends to A, and a plane (AMN) would converge to the tangent plane as M and N tend to A along curves drawn on the surface that are not tangent to each other at A.
Does this answer your question?