Answer: C

Small Hint:

Figures $A$ and $B$ leave the same shaded area.

Big Hint:

For $C,$ compare $\pi-2$ with $4-\pi.$

Solution:

In figure $A,$ the shaded area is the area of a $2$ by $2$ square minus a circle of radius $1,$ so it is $4-\pi.$

Figure $B$ is made of four copies of figure $A$ with half the side length, so each copy has one-fourth the area and the total shaded area is also $4-\pi.$

In figure $C,$ the circle has radius $1,$ and the inscribed square has diagonal $2,$ so its area is $2.$ The shaded area is $\pi-2.$

Since $\pi>3,$ we have $\pi-2>4-\pi,$ so figure $C$ has the largest shaded area.

Thus, C is the correct answer.