Question

Find the area of the shaded region. Thanks!

Answer 1

Check the picture below.

now, let's rationalize the denominator for "r" first.

also notice, that section is just 1/4 of the area of that circle, so, let's just find the area of the whole circle with that "r", get one-quarter of it, and subtract it from the half of the area of the square.

$\bf r=\cfrac{8}{\sqrt{2}}\cdot \cfrac{\sqrt{2}}{\sqrt{2}}\implies r=\cfrac{8\sqrt{2}}{2}\implies \boxed{r=4\sqrt{2}} \\\\\\ \stackrel{\textit{area of that circle}}{A=\pi (4\sqrt{2})^2}\implies A=\pi (16\cdot 2)\implies A=32\pi \\\\\\ \textit{how much is one-quarter of that?}\quad \cfrac{32\pi }{4}\implies \boxed{8\pi }\\\\ -------------------------------$

$\bf \stackrel{\textit{area of the square}}{A=8\cdot 8}\implies A=64\qquad \stackrel{\textit{half of that square's area}}{32}\\\\ -------------------------------\\\\ \textit{shaded region}\implies \stackrel{\stackrel{\textit{half of that square}}{area}}{32}~~-~~\stackrel{\stackrel{\textit{one-quarter of the circle}}{area}}{8\pi }$