THE PROBLEM : In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way.
Let p be an interior point of the disk, and let n be a multiple of 4 and greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of 2π/n radians, and slicing the disk on each of the resulting n/2 lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that:
The sum of the areas of the odd-numbered sectors equals the sum of the areas of the even-numbered sectors .
The pizza theorem is so called because it mimics a traditional pizza slicing technique. It shows that, if two people share a pizza sliced in this way by taking alternating slices, then they each get an equal amount of pizza.
QUESTION I : Could there be a proof of this theorem using calculus?
QUESTION II : Could this theorem be generalized to sphere?
REFERENCE : Pizza Theorem – Wikipedia