The problem asks for the area of the blue region in the diagram, where $AC = BC$ and the radius of the semicircle is 4 units. To solve this, we need to break the problem into simpler parts.

Step 1: Calculate the area of the semicircle.

The area $A_{\text{semicircle}}$ of a semicircle is given by:

$A_{\text{semicircle}} = \frac{1}{2} \pi r^2$

Since the radius $r$ is 4 units:

$A_{\text{semicircle}} = \frac{1}{2} \pi (4)^2 = \frac{1}{2} \pi (16) = 8\pi \text{ square units.}$

Step 2: Calculate the area of the triangle $\Delta ABC$.

The triangle $ABC$ is an isosceles triangle with base $AB = 8$ units (since $AO = 4$ and $OB = 4$) and height $OC = 4$ units (which is the radius of the semicircle). The area of the triangle is:

$A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 4 = 16 \text{ square units.}$

Step 3: Subtract the area of the triangle from the area of the semicircle.

The blue region is the area of the semicircle minus the area of the triangle:

$A_{\text{blue region}} = A_{\text{semicircle}} - A_{\text{triangle}} = 8\pi - 16 \text{ square units.}$

Step 4: Conclusion

The correct formula for the area of the blue region is $8\pi - 16$, and the closest match to the given options would likely involve simplifications depending on how the answers are presented.

Would you like to further clarify or ask anything else? Here are some related questions:

1. How is the formula for the area of a circle derived?
2. Why does subtracting the triangle area give us the blue region?
3. What is the difference between the full circle and semicircle area calculations?
4. How does symmetry in geometry help in solving problems like this?
5. What is the significance of recognizing this as an isosceles triangle?

Tip: Always double-check units in geometry problems to ensure consistent results!