Answer
$2\pi-4\quad$ square units.
Work Step by Step
The area of the shaded region is the difference between the area of the circle and the area of the square.
Thus, we first need to find the area of the circle and the area of the square.
The side length of the square is
$s=2$.
Area (let this be $A_1$) of the square is
$A_1=s^2$
$A_1=2^2$
$A_1=4\quad$ square units
To find the area of the circle, we need to find its diameter which is actually the diagonal of the square.
Note that length of the diagonal of a square with sidelength $s$ is $s\sqrt2$.
Thus, the length of the diagonal of the square (which is the diameter of the circle) is
$=s\sqrt2$
$=2\sqrt2$
Hence, $d=2\sqrt2$.
The area of the circle (let this be $A_2$) is:
$A_2=\pi \left(\frac{d}{2}\right)^2$
$A_2=\pi \left(\frac{2\sqrt2}{2}\right)^2$
$A_2=\pi(2)$
$A_2=2\pi\quad$ square units
Therefore, the area $A$ of the shaded region is
$A=A_2-A_1$
$A=2\pi-4 \quad$ square units
Hence, the area of the shaded region is $2\pi-4$ square units.
