The diagonal of the square is : 4 x 2 = 8 (in)

The edge of the square is : \(\dfrac{8}{\sqrt{2}}=4\sqrt{2}\) (in)

The area of the square is : \(\left(4\sqrt{2}\right)^2=32\) (in²)

The area of the large circle is : \(8^2\pi=64\pi\) (in²)

The area of the small circle is ; \(4^2\pi=16\pi\) (in²)

The answer is : \(64\pi-16\pi+32=48\pi+32\) (in²)

We easy see the diagonal of the square is diameter of smaller circle.

=> The diagonal of the square is: 4.2 = 8 [in]

=> The edge of the square is: \(\dfrac{8}{\sqrt{2}}=4\sqrt{2}\) [in]

=> The area of square is: \(\left(4\sqrt{2}\right)^2=32\left(in^2\right)\)

The area of smaller square is: 4².3,14 = 50,24 [in²]

The area of bigger square is: 8².3,14 = 200,96 [in²]

=> The total area of the shaded regions is:

\(\left(200,96-50,24\right)+32=182,72\left(in^2\right)\)

\(182,72\approx58,16\pi\)

So the answer is 58,16\(\pi\)