Let's analyze the problem step by step.

The figure shows a circle with a shaded region. The square with side length 4 units is inscribed in the circle, and the shaded region is the portion inside the circle, but outside the square. The task is to calculate the area of this shaded region.

Step 1: Find the area of the circle

The circle is centered at point O, and the diameter of the circle is equal to the diagonal of the square. Since the side length of the square is 4, we can use the Pythagorean theorem to find the diagonal (d) of the square.

$d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

The radius (r) of the circle is half of the diagonal:

$r = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$

The area of the circle (A_circle) is given by:

$A_{circle} = \pi r^2 = \pi (2\sqrt{2})^2 = \pi \times 8 = 8\pi$

Step 2: Find the area of the square

The area of the square (A_square) is:

$A_{square} = \text{side}^2 = 4^2 = 16$

Step 3: Find the area of the shaded region

The shaded area is the area inside the circle minus the area of the square:

$A_{shaded} = A_{circle} - A_{square} = 8\pi - 16$

Conclusion:

The area of the shaded region is $8\pi - 16$, which is the answer in the context of this problem.

The closest match from the options is:

A) 8π

Would you like a further explanation or have any additional questions about the problem?

Here are some related questions:

1. How do you calculate the diagonal of a square?
2. What is the formula to calculate the area of a circle?
3. How do you find the area of a shaded region between two shapes?
4. What is the Pythagorean theorem, and how is it applied in geometry?
5. How does the radius of a circle relate to its diameter?

Tip: When working with geometric figures, always identify known values (like side lengths or radii) and apply formulas to find unknown quantities step by step.