Actually, the solution is not wrong, it's just that you are thinking of another kind of problem.

The reason this is called "Pizza Cut" is that a pizza is flat.  Yes, technically it is three dimensional (or else you would never get full), but for all intents and purposes, it is like a 2D circle.

To solve the problem you are talking about, see Partitioning the Regions, which discusses a "cheesecake cut" with deals with a three dimensional cylinder.