jmc

First we have to find the area of the shaded region in each of the figures. In figure $\text{A}$ the area of the shaded region is the area of the circle subtracted from the area of the square. That is $2^2-1^2\pi =4-\pi$. In figure $\text{B}$ the area of the shaded region is the sum of the areas of the 4 circles subtracted from the area of the square. That is $2^2-4\left({\left(\frac{1}{2}\right)}^2\pi \right)=4-4\left(\frac{\pi }{4}\right)=4-\pi$. In figure $\text{C}$ the area of the shaded region is the area of the square subtracted from the area of the circle. The diameter of the circle and the diagonal of the square are equal to 2. We can easily find the area of the square using the area formula $\frac{d_1{d}_2}{2}$. So the area of the shaded region is $1^2\pi -\frac{2\cdot 2}{2}=\pi -2$. Clearly the largest area that we found among the three shaded regions is $\pi -2$. Thus the answer is $\boxed{\text{C only}}$. * Note - If you don't know the diagonal formula, you can also find the length of the square using the Pythagorean Theorem (set up an equation and solve for $x$), which is $\sqrt{2}$, and get that the area is $2$, and continute as shown in the solution.