Have you heard of Aristotle's wheel paradox?

The inner circle does have a circumference. Pull it out from the bigger wheel and it will turn and travel a different distance from the bigger wheel.

Even as part of the big wheel the small wheel will turn and not be “dragged”. Why? Because the small wheel IS the big wheel.

The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length are all the same, so the points of any of these can be put in a one-to-one correspondence with those of any other.

That’s not a paradox; it’s a geometry mistake.

It is implying that r1 * theta = r2 * theta, which is incorrect. Assuming r1 < r2: If you have a string at r1 and one at r2, when you roll on r2 to wrap the string around the wheel, the wheel will shift by 2Pi * r2 and the string on r1 will droop. It won’t wrap nicely as they are implying. Thus no paradox.