I have noticed Can I Haz My Eye Center'd? puzzle, which I know in slightly different formulation:

You have a disk on a table and a point A on it. You need to cut the disk into smallest possible amount of pieces and then glue them back into a disk, which has centre at A. But: 1. you are not allowed to take pieces out of the table plane (you can only slide them). 2. you need to use as little glue as possible (e.i. the cut length should be minimal).

Henning Makholm has already shown a way in his answer here, which allows to cut into the smallest amount of pieces, but the cut length still can be optimised and once you do it you will see this puzzle differently.