The other day, I was ordering pizza with someone, and I came across an interesting property. You see, we both wanted the same type of pizza, and each knew what size we wanted individually, so we decided to sum the areas of each of our desired pizzas and get one together that has an equal or greater area (so we got the amount of pizza each of us wanted).
To find the radius of the pizza we want, R, given the radii of each pizza individually, $r_1$ and $r_2$ and the formula for the area of a circle, $A=\pi r^2$:
$$\pi (r_1^2) + \pi (r_2^2) = \pi (R^2)$$
Dividing by $\pi$ yields an all too familiar theorem from Algebra I.
$$ r_1^2 + r_2^2 = R^2$$
My question is – how are these related? What does this relationship between adding the areas of two circles and the relationship between side lengths of a right triangle mean geometrically? It's entirely possible that this is a coincidence, but, I figured, maybe someone out there smarter than I can synthesize this correlation into something meaningful.
Best Answer
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
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.