Are circles related to the sum of exterior angles of convex polygons equalling 360 degrees?

Question

The sum of all the exterior angles of a convex polygon equal 360 degrees, and a full circle is equal to 360 degrees. The more sides you add to a convex polygon, the closer you get to a circle.

So do circles have anything to do with why a convex polygon's exterior angles will always add up to 360 degrees?

Answer 1

Not in the sense you are suggesting, as there is no reason for the limiting convex polygon to be circular, no matter how many sides you have.

But yes, in the sense that the external angles add up to a rotation around a full circle if you shrink the polygon down to a point, as in