A square of side length $2$ and a circle of radius $\dfrac{2\sqrt3}3$ share the same center. What is the area inside the circle but outside the square?

1. With square side $s$ and circle radius $r$ (so $r>\frac s2$), each of the $4$ circular segments bulging past a side has central angle $\theta=2\arccos\!\left(\frac{s/2}r\right)$, and the total shaded area is $2r^2(\theta-\sin\theta)$.
2. With $s=2,r=\frac{2\sqrt3}3$: $\frac{s/2}r=\frac{1}{2\sqrt3/3}=\frac{\sqrt3}2$, so $\theta=2\cdot30^\circ=60^\circ=\frac\pi3$.
3. Area $=2\cdot\frac43\left(\frac\pi3-\frac{\sqrt3}2\right)=\boxed{\dfrac{8\pi}9-\dfrac{4\sqrt3}3}$.