jmc

The area of the smaller square is one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter: $2\cdot 2\cdot \frac{1}{2}=2.$ The circle's shaded area is the area of the smaller square subtracted from the area of the circle: $\pi -2.$ If you draw the diagonals of the smaller square, you will see that the larger square is split $4$ congruent half-shaded squares. The area between the squares is equal to the area of the smaller square: $2.$ Approximating $\pi$ to $3.14,$ the ratio of the circle's shaded area to the area between the two squares is about $$\frac{\pi -2}{2}\approx \frac{3.14-2}{2}=\frac{1.14}{2}\approx \boxed{\frac{1}{2}}$$