What proportion of the quarter circle is shaded? 
Interesting yet challenging quiz I found on a website. My answer is a $\frac{1}{ \sqrt{2}}$.
After I assumed the semicircle has radius $r\sin{45}$, where $r$ is the radius of the quarter circular part.
Any objections or comment?

Let $R$ be the radius of the outer circle and  $M=(r,r)$ be the center of the brown semidisc. Then $|OM|=\sqrt{2} r$ and therefore $R^2=3r^2$. The ratio of the areas then comes to ${2\over3}$.
