UKMT Intermediate Mathematical Challenge, 2017

The diagram shows a circle with radius $2$ and a square with sides of length $2$. The centre of the circle lies on the perpendicular bisector of a side of the square, at a distance $x$ from the side, as shown. The shaded region − inside the square but outside the circle − has area $2$. What is the value of $x$?

(A) $\dfrac{\pi}{3} + \dfrac{\sqrt{3}}{2} - 1$ (B) $\dfrac{\pi}{3} + \dfrac{\sqrt{3}}{4} - 1$ (C) $\dfrac{\pi}{3} + \dfrac{1}{2}$ (D) $\dfrac{\pi}{3} + 1$ (E) $\dfrac{\pi}{3}$