Zeta 137: 20. Pizza

11/04/2026

20. Pizza

The beach is a good place to reflect. A parasol reminded me of an elementary geometry theorem that proves the equality of two areas obtained by properly partitioning a circle: the pizza theorem, if a circular pizza is divided into 8, 12, 16, ... slices by making equiangular cuts, the sum of the areas of the even slices is equal to that of the remaining odd slices.

First Theorem. If two people cut a pizza into 4n + 4 equiangular sectors (with whole n greater than zero), centered at any point, and take turns taking a slice each, walking the pizza clockwise or counterclockwise, both will eat the same amount. In addition, sectors can also be grouped into n+1 equivalent in area sets. So, e.g. a pizza cut into 8 slices can be divided equally between 2 people, one divided into 12 slices between 3 and one divided into 20 between 5.

Now let's look at some other theorems (to be taken more or less seriously ...).

Second Theorem. The total area of the green slices is equal to that of the red slices (regardless of where P is placed). In this case, it is not the corner in the center, but the edge of each slice that is kept constant. Note: it is easy to verify that the same theorem applies to the figure on the right, which shows a regular polygon instead of a circle, but in this case the proof is simpler.

Third theorem. If you divide the pizza into 6 parts (60-degree angles), the following equality always applies:

Fourth theorem. The Pythagorean theorem also applies to pizza: the area of the pizza built on the hypotenuse is equal to the sum of the areas of the pizzas built on the catheti:

Fifth Theorem. Indicating with Pi the well-known 3.14, the volume of a pizza with radius z and thickness a is given by the expression:

Pi x z x z x a

Sixth Theorem. You can divide a square pizza equally between two people by cutting it four times: vertical, horizontal and diagonal cut (at 45 degrees) by placing the crossing point anywhere within the square: