Center of wheel travels the length of circumference in one revolution I was wondering if there is a more mathematical/rigorous way of seeing that the wheel/circle/its center travels the length of wheel's circumference in one revolution.
Intuitively, one could cover the wheel/circle with a string the length of which is exactly equal to its circumference. Then in one revolution the string would be spread so that we can see the center traveled the length which is equal to the circle's circumference.
 A: What you described is not a rule, it's only the case when you have rolling without slipping.
These two lengths you've described are usually independent on each other.
A: Special case
When a Circle radius R ( circumference = $ 2 \pi R $) rolls on a straight line without slipping, a point on its rim traces out a cycloid when the length traversed by the Circle center is also the same$=2 \pi R$.
A: In the left hand diagram the wheel is rolling along the ground, point $A$ is in contact with the ground and has zero speed.  For small changes in time it's as if the green line is being tilted, pivoted around point $A$,  so if point $O$ has a speed of $v$ m/s, then point $B$ has a speed of $2v$ m/s.

In the right hand diagram the wheel is shown from the point of view of an observer travelling with the wheel.  Point $O$ is now stationary, point $A$ moves left at $v$ m/s and  point $B$ moves right at $v$ m/s.
If the observer watches a red spot of paint on the rim it takes time $$t=\frac{2 \pi r}{v}$$ to return to the bottom, the wheel has then made a complete turn.
In this time, as seen from the left hand diagram, the wheel would travel a distance $$vt = \frac{2 \pi rv}{v} = 2 \pi r$$
