@joe-shmo said
A circle is inscribed in a square of side length 2. The region bound between the square and the circle is shaded. A copy of that figure is inscribed in the circle that is inscribed in the original square, and another copy inscribed in that circle ad infinitum...
What is the total shaded area?
These things are always fun, because at first glance it looks like you're going to have to go into an infinite regression and use differential equations or something like that, but if you look at it right you see that you'll just end up with a single equation where the same ratio figures on both sides.
In this case, the area of the square is 4 and that of the circle inside is π. The shaded area outside the largest circle is, therefore, 4-π. And the total shaded area (let us call that, simply, S) is that, plus the shaded area
inside the circle... but since the proportions scale, that latter one is the same as S times the size of the second square divided by the larger one (i.e. 4).
So, S = 4-π + S * proportion; or, S = (4-π ) / (1-proportion[exscribed square/inscribed square]).
Now all that's left to do is find that proportion. Have patience, please, while I try to recall my trig...