It is well-known that a disc can't be cut into finitely many pieces and reassembled into a square of the same area. This is Tarski's famous problem, except only allowing Jordan curve cuts. It seems obvious to me that in attempting to do so, the exposed convex edges of the perimeter of the circle can only be eliminated, by cutting corresponding segments of concave curves to fit against it. But so doing is guaranteed to generate new off-cuts with new edges having equally convex curves of total length $2\pi$.
Therefore the $2\pi$ length convex curve of the circle's perimeter can never be eliminated. This is a simple proof which has always satisfied me. How difficult is it to turn this into a rigorous proof?
I can imagine that we would start by defining the set of all exposed edges (initially the perimeter) and then prove that for any set of cuts, there will always be at least length $2\pi$ of exposed edges having curvature equal to the original circle. Since squares have only straight exposed edges, the proof is complete.
Surely this can be a simple, elegant and obvious proof of something that for a long time was not accepted by many?