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Hypothetically asking, let's say there is a circle with an unknown radius. Inside this certain circle, there is a square. The square's four corners touch the circle's circumference. If you remove the square what percentage of the entire circle would be left.

If my explanation was not clear enough, I have made a diagram demonstrating what I am asking: http://s.codepen.io/PartTimeCoder/debug/LRXZPG

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Suppose the circle is a unit circle (a circle of unit radius); its area is then $\pi r^2 = \pi$.

Then the diagonal of the square is $2$, which means that its side is $\sqrt{2}$, which means that its area is $2$.

So the remaining area is $\pi-2$, and the fraction of area remaining is $\frac{\pi-2}{\pi}$.


ETA: You can convince yourself, by using circles of radius $k$ and squares of diagonal $2k$, that the scale of the circle and square does not matter; the percentage remains the same.

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  • $\begingroup$ Would the percentage remaining be 36%? $\endgroup$
    – user380564
    Oct 19 '16 at 23:47
  • $\begingroup$ @AnishKasam: Yes, approximately. $\endgroup$
    – Brian Tung
    Oct 20 '16 at 16:36

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