Informal descriptions of the Ehrenfest “paradox” typically start with the mention of a rigid disc or cylinder. But STOP right there! In special relativity nothing is classically rigid as a truly rigid body would instantly transmit acceleration from one point to another, which contradicts the impossibility of sending a signal faster than the speed of light.
Ehrenfest’s actual observation was not intended as pointing out a logical inconsistency in special relativity but just to point out the limited applicability of one early attempt to identify the kinds of accelerated motion which could preserve distances between a set of moving points. This quickly gets too technical for a Quora answer but the Wikipedia articles on Born Rigidity and Ehrenfest Paradox provide a basic introduction to the relevant literature.
Even without bringing in rigidity though, it can be interesting to ask what special relativity says(*) about how various observers will perceive a rotating disc or wheel – which, for comparison purposes we will place directly beside a non rotating wheel of the same size.
*Aside: It is important to note here that bringing in General Relativity “because … acceleration” is a complete red herring. The question is not about what “really” happens but about whether what SR predicts is internally consistent. And SR certainly does make predictions about accelerated motion and about what will be seen by accelerated observers.
If someone on the rim of each wheel goes slowly around and measures off equal distances around the circumference, then from the point of view of an observer who is at (or not moving relative to) the centre, since the spinning wheel rider’s ruler appears contracted, the central observer will see more markings on the rim of the spinning wheel than on that of the stationary one. But the number of markings is an objective fact so the moving observer must agree – even though every time she looks at the interval at her feet she sees the one next to it on the stationary wheel as seeming shorter. How can this be?
Well let’s look at what the spinning wheel rider sees of the situation as a whole.
In particular note that as she looks at the far side of both wheels, although she sees the distant interval on the stationary wheel as shorter than that at her feet, she also sees the far side of her own wheel as moving twice as fast, so the markings she previously made over there will now appear to have an even greater contraction. And of course everything must work out so that when every part of both wheels is taken into account she comes up with the same numbers of markings as the stationary observer. But it would be undermining a nice homework exercise for an undergrad SR course if I were to work it all out right here.
P.S. Also, for an observer on the spinning wheel, both wheels appear contracted in her tangential direction (which keeps changing so the wheels certainly don’t strike her as “rigid”) and so I think we can stop worrying about whether she gets a different value for [latex]\pi[/latex].
P.P.S. With regard to the [latex]\pi[/latex] question, one might forget about the wheels and ask what the rotating observer will make of the fact that she is constantly accelerating towards the centre of her wheel and how she will relate the radius and circumference of her actual path (which she will infer is circular, though not coinciding with her current perception of the rest of the wheel). The “solution” in this case is that she will not see the centres of the wheels as fixed at the centre of her path but rather as always on the near side of it and rotating around it with her so that the radius she gets for her path is greater than that which the stationary observer gets for the wheels. (This effect can be analysed in terms of the limiting case of many successive small “frame jumps” of the same kind that are used to resolve the twin paradox.)