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.

When you want to outsmart the world, you turn to the facts. And the facts are in the science.