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.

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    $\begingroup$ It is not related to the circle. If all the pizzas were homothetic to a different shape, the same rule would apply. $\endgroup$ – Gribouillis Dec 26 '19 at 7:07

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

  1. There are two figures whose areas should sum to a third
  2. 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.

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    $\begingroup$ This is the right answer here. It's not about the Pythagorean theorem per se. The pattern is "two areas add up to a third area", a concept that applies to pizza sizes and to the Pythagorean theorem. $\endgroup$ – Tim Pederick Dec 26 '19 at 17:01

Notice that in the following diagram, the areas of the smaller squares add up to the area of the larger square.

enter image description here

However, the shape doesn't actually matter. You could make them semicircles (and therefore the circle relationship follows easily) or pentagons or anything else.

enter image description here

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    $\begingroup$ I will note that this leads to the most (imo) conceptually satisfying proof of the pythagorean theorem: draw the altitude from the right angle vertex to the hypotenuse. This splits the triangle into two similar triangles. Since "the shape doesn't matter", you see that you could use squares instead of triangles, and the pythagorean theorem follows. $\endgroup$ – Steven Gubkin Dec 26 '19 at 15:11
  • $\begingroup$ @StevenGubkin or simply measure the shape by its diameter, which in this case is the length of the hypothenuse. $\endgroup$ – Gribouillis Dec 26 '19 at 22:00

This is a proof without words of the Pythagorean theorem. As long as I know, triangles aren’t inherently (in some sense) related to circles, but you can see that areas can be used to have an analogy to the figures.


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