The Pizza Theorem. Weekly Problem no. 8

Problem:

A pizza is cut by 8 straight lines through a point P, which is not at the center of the pizza. The cuts are made at equal angles (45° apart). Prove that the sum of the areas of alternate pieces is equal.

Questions to Solve:

1. If point P is 2 cm from the center of a pizza with radius 10 cm, what is the largest possible difference between any two adjacent pieces?
2. How does moving point P affect the areas of individual slices?
3. Why does the theorem still work even when P is not at the center?

Solution Approach:

1. First, understand that:

• The 8 cuts create 16 pieces
• The odd-numbered pieces sum to the same area as the even-numbered pieces
• This is true regardless of where point P is located (as long as it’s inside the pizza)

2. Key insights:

• Equal angles at P ensure proportional division of any circle centered at P
• The pizza can be viewed as an infinite sum of circular strips
• Each strip is divided proportionally, leading to equal alternating sums