What is the area of the region inside the circle $x^{2} + y^{2} = 25$ and outside the square $\lvert x \rvert + \lvert y \rvert = 5$?

Explanation

1. 01

   In the first quadrant the side of the square is the chord $x + y = 5$; the piece there is $\int_0^{5} \left(\sqrt{25 - x^{2}} - (5 - x)\right)dx = \frac{25\pi}{4} - \frac{25}{2}$.

2. 02

   The four quadrants give equal pieces: $4\left(\frac{25\pi}{4} - \frac{25}{2}\right) = 25\pi - 50$ square units.