This isn't quite right; when you divide up your problem, your "quarter circle" doesn't look the same after each iteration, and so the sizes of the squares cut off don't follow a geometric sequence. In fact, I'm sure that the area of the figure does approximate the area of a circle (because no point inside the circle is ever removed, yet every point outside of it will be removed eventually).
The problem lies not with the area, but the perimeter; the figure's area turns out to be unrelated for this problem.
Hmh. Its 1 AM here. Give me some time to sleep and think over that. You might be right: I can't remember why I thought the squares would go geometric. I thought I had a better reason than "It looks that way on the picture, kinda...", but it eludes me.
If A_n is a sequence of sets bounded by these square like curves, then m(A_n) -> pi, where m is a Lebesgue measure. Each set is clearly measurable, since it is closed in Euclidean topology, and thus Borel. You can prove this sequence converges by seeing that each A_n is enclosed in some circle of diameter 1+e for some e, and, as n -> \infty, e -> 0.
Right. This way, you aren’t approximating the circle. And if you were, the area would converge to the right number. Areas are good this way, unlike lengths which is what the post is about.
The problem lies not with the area, but the perimeter; the figure's area turns out to be unrelated for this problem.