The square has a side length equal to the diameter of the circle: $2 \times 4 = 8$.

$\text{Area of square} = 8^2 = 64$

$\text{Area of circle} = \pi(4)^2 = 16\pi$

The shaded region is the area inside the square but outside the circle:

$\text{Shaded area} = 64 - 16\pi$