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I'm a mathematician, and I actually hadn't seen the proof that $A = \pi r^2$ via showing that the shape formed by that particular arrangement of the slices of the circle becomes a square as the number of slices is taken to infinity. That's a very cool proof.

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Me too. I love it when a non-trivial concept like that can be effortlessly explained with a good diagram.

Out of curiosity, were you perhaps taught with some other seemingly-intuitive explanation? The one given here was the first (and, really, only) such explanation that I received in grade school.

It was also my first real exposure to the concept of infinity and limits in math class (although my brother had stumbled upon and explained to me the concept of a derivative in his head one day while sitting in church, way before either of us would be exposed to calculus).

He didn't prove the wiggly-sided paralellogram -> rectangle transition when the angle of each slice goes to 0. He also did not prove that he is allowed to take the angle to 0. I can grant him that any angle > 0 results in a less and less squigglier paralellogram with more and more vertical sides with an obvious proof, but that doesn't prove either that the limit is a rectangle, or that reasoning at the limit is transitive to reasoning about the circle.

I don't doubt the result or the method - they both seem intuitively correct, they just aren't proven.

Good luck seeking proof.

http://en.m.wikipedia.org/wiki/Regress_argument

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