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.