An Unconventional Proof of the Pythagorean Theorem Using Circles

Question

In this article (its a short read)...

https://betterexplained.com/articles/pythagorean-sweep/

...the author tries to intuitively show the Pythagorean theorem using circles.

However, there's one part that confuses me. The intuition seems to entirely rely on us being able to "see" that in this picture...

...the area swept out by "line B" is $\pi B^2$.

Is there a way to intuitively show this?

Thanks!

Answer 1

The author doesn't use circles to prove the Pythagorean theorem. Rather, they use the Pythagorean theorem to prove a statement involving circles. By the Pythagorean theorem, the radius of the circle swept out by Lines A and B in the cited diagram is $\sqrt{a^2+b^2}$, so its area is $\pi(a^2+b^2)$. The area of the circle swept out by Line A is $\pi a^2$. So the area of the ring is $\pi b^2$.

However, this idea does give rise to an alternate proof of the Pythagorean theorem as follows. Let $c$ be the radius of the circle swept out by the perpendicular Lines A and B. If we extend Line B to a chord of length $2b$, then Euclid's Proposition III.35 tells us that $(c+a)(c-a)=b^2$, that is, $a^2+b^2=c^2$.

$\hspace{2 cm}$

But this isn't what the author of the linked article does.