The Pizza Theorem Will Make You Look Great

The pizza theorem is the perfect topic to bring up during an evening with friends. It's a fun way to pass the time while talking about something interesting because everyone enjoys it, and it's also a great resource for those times when the conversation takes an awkward turn (which is no small feat). In no time, I assure you, you'll have the entire table's attention.

What is the pizza theorem for?

Essentially, it helps you figure out how to divide your capricciosa, margherita, or any other type of pizza into equal parts. Or perhaps a pie, both sweet and savory, or any other round dish you might find on the table. "Pizza theorem," in fact, is a nickname: the simplification of a mathematical puzzle dating back to the late 1960s.

How the Pizza Theorem Was Born

It's 1968, and L.J. Upton posed a problem in Mathematical Magazine. The text read: "Four lines in a plane all pass through the same point O. The angles between the lines are all 45°. A circle is superimposed on this configuration so that O falls inside the circle. (a) Show that the alternate sectors cover half the area of the circle. (b) Prove the result without using calculus."

How to solve Upton's problem? Michael Goldberg, also a mathematician, suggests dividing the disk into four (and multiples of four) equiangular sectors. Shortly thereafter, Larry Carter and Stan Wagon provide a graphical demonstration: the following, considered the most effective representation of the theorem, is actually more difficult to explain in words.

Based on this scheme, other mathematicians have expanded on the theory, explaining that the circle could also be divided by three, and so on, with other multiples. But for us, the Carter and Wagon scheme is sufficient, which essentially states this: a circle (like that of a pizza, for example) can be divided into two equal parts without passing the dividing line through the center, but only if it respects the rule of multiples of 4.

How to apply the pizza theorem

To oversimplify: take the pizza and make a cut along its length, starting from any point you like. Then make another perpendicular cut, forming four 90-degree angles. Divide these angles again to form eight, then a 45-degree cut, making another cut that passes through exactly the same point as the others. Now, to ensure you're eating exactly the same amount of pizza, take it in alternating slices, always proceeding in the same direction: clockwise or counterclockwise. Here's another example:

The point is, to be as precise as a mathematician, pizza should be perfectly circular, and this doesn't always happen. In fact, almost never. Perfectly circular pizzas are more common in the United States or elsewhere. And they're also much larger than ours, designed for a different consumption. Which is why knowing exactly how to divide them so as not to displease anyone, trying an alternative and fun method, makes more sense. But anyway, at the next dinner we can still say we know the pizza theorem. That's no small feat.