Area of shaded region

Since the radius is equal to $\frac {3 \sqrt{2}}{2}$ , we can deduce that the side length of the square is 3. So the area of the shaded region equals the area of the square minus the area of the two right triangles not shaded. The area of the square is obvious 9, and we can easily find both triangles' areas: $\frac{3}{2} \cdot 3 = \frac{9}{2}$ . And $9-\frac{9}{2} = \frac{9}{2} = 4.5$ .

The diagonal of the square is also the diameter of the circle. The diagonal of the square is given by $d=x\sqrt{2}$ . Substituting, we get

$3\sqrt{2}=x\sqrt{2}$

$x=3$

So $AE=\dfrac{3}{2}=1.5$ .

The unshaded part inside the square is a $3 \times 1.5$ rectangle. The area of the shaded part is equal to the area of the square minus the area of the unshaded part.

$3^2-(3\times 1.5)=9 - 4.5 = \boxed{4.5}$

Let the side length of the square be $x$

By Pythagoras, $x^2 + x^2 = (3\sqrt{2})^2 = 18$ , therefore $x=3$

Square Area = Shaded Area + Non-shaded Area

$\implies$ Shaded Area = Square Area - Non-shaded Area

$= 9 - \triangle AED - \triangle CFD$

$= 9 - 2.25 - 2.25 = 4.5$