Relation between adding the areas of two circles and Pythagorean Theorem
If the pizza shop sold square pizzas, where the size of the pizza was expressed as a function of the length of a side, then your relationship would be $$s_1^2 + s_2^2 = s_3^2,$$ where $s_1$, $s_2$ are the side lengths of the individual pizzas, and $s_3$ is the side length of the pizza to be shared.
So, as you can see, this is not a relationship peculiar to a circle. It is simply a consequence of the fact that
- There are two figures whose areas should sum to a third
- The area is proportional to the square of some linear dimension of the figure.
If you were to do this for three people wishing to combine their order for circular pizzas (or square, or rectangular, etc.), you'd get the sum of three squares equaling a fourth.
Notice that in the following diagram, the areas of the smaller squares add up to the area of the larger square.
However, the shape doesn't actually matter. You could make them semicircles (and therefore the circle relationship follows easily) or pentagons or anything else.
