How do you find the area of the shaded area of this circle? 
I need to find the area of the shaded area. The triangle is equilateral. So far, I have found the area of the triangle to be $\sqrt 3$, but I cannot figure out how to find the radius of the circle in order to find the area of the circle. Any advice would be appreciated.

We're looking for the radius right? So let's draw them..

Please excuse the drawing.
Ok. Now we have an isosceles triangle $30-30-120$. If  $r$ is the radius then law of sines tells us,
$$\frac{2}{\sin 120}=\frac{r}{\sin 30}$$
So $r=2 \frac{\sin 30}{\sin 120}=2\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}$. I think you can take it from here.
