I think that is because as the number of teeth increase the polygon we're describing more closely approximates to a circle. A circle has a constant ratio of circumference to diameter - pi. Whereas, for the regular decagon of a 10 tooth sprocket, the ratio of the diameter and perimeter is 3.09, for a hexagon it is 3, and the ratio of the diagonal of a square to the perimeter is 2 time square root of 2 or 2.828. The difference between each difference will get smaller and smaller.
Had to read that about ten times, but I see what you're saying.
I just thought, the size of each additional tooth is constant - whereas the circle is constantly getting bigger. In my head, that means the bigger the 'circle' gets, the less one additional tooth would stretch it. But obviously not.
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