FabR[Senior Patek Moderator]
26166
Solution (just including for your son the details of what our friend Guycord wrote, which was both clear and correct): The shaded area is $32 \pi - 48$. If you approximate $\pi $ by 3.14, then this area is roughly 52.48.
May 08, 2022,14:04 PM
We just use that the area of a circle of radius $R$ is $\pi R^2$, hence the area of each quadrant is $\pi 8^2 / 4=16 \pi$. Also, by Pythagoras' theorem, the diagonal of a square is $\sqrt{2}$ times its side (where \sqrt indicates the square root function). This gives that the side of the larger square is $8 / \sqrt{2}$, and therefore its area is of course $(8 / \sqrt{2})^2=64 / 2=32$. Finally note that, by symmetry, the smaller square has area one half of the larger one, or 16.
Put the above together to conclude for instance that the total shaded area is the sum of the two quadrants, minus twice the larger square, plus the smaller square: $2 x 16\pi - 2 x 32 + 16=32 \pi - 48$.
Since I did this in the middle of the summer when I'm not even under academic contract, in return tell your son that he owes me to become a HUGE Patek collector!ππβ€οΈ