The problem asks to find the area of the shaded region of a circle. The circle has a radius of $4$ cm, and the central angle of the unshaded sector is $90$ degrees. The answer should be in terms of $\pi$ and should include the correct unit.

1. Problem Description

The problem asks to find the area of the shaded region of a circle. The circle has a radius of $4$ cm, and the central angle of the unshaded sector is $90$ degrees. The answer should be in terms of $\pi$ and should include the correct unit.

2. Solution Steps

First, we calculate the area of the entire circle.

The area of a circle is given by the formula:

$A_{circle} = \pi r^2$

where $r$ is the radius of the circle. In this case, the radius is $4$ cm.

$A_{circle} = \pi (4 \text{ cm})^2 = 16\pi \text{ cm}^2$

Next, we calculate the area of the unshaded sector. The central angle of the sector is $90$ degrees, which is $\frac{90}{360} = \frac{1}{4}$ of the entire circle.

The area of the sector is given by:

$A_{sector} = \frac{\text{central angle}}{360} \times A_{circle}$

$A_{sector} = \frac{90}{360} \times 16\pi \text{ cm}^2 = \frac{1}{4} \times 16\pi \text{ cm}^2 = 4\pi \text{ cm}^2$

Now, we subtract the area of the unshaded sector from the area of the entire circle to find the area of the shaded region.

$A_{shaded} = A_{circle} - A_{sector}$

$A_{shaded} = 16\pi \text{ cm}^2 - 4\pi \text{ cm}^2 = 12\pi \text{ cm}^2$

3. Final Answer

$12\pi \text{ cm}^2$