The exact answer to this question comes from deep in the field of mathematical analysis, which is where mathematicians go to think up worse-case scenarios. This zig-zag path does approach the circle, so on the surface of it you’d expect that it would have the same length. However, the length of a curve is more a function of it’s derivative (slope), and less a function of its position. If you inscribe an N-gon (an N-sided polygon) into a circle, the length of it’s perimeter is:, where D is the diameter of the circle. The inside the Sine function falls out of the definition of “radians“, and basically boils down to the usual definition.