The problem asks us to find the area of the shaded region of a circle. The circle has center $O$ and radius of $4$ cm. The central angle of the unshaded region is $90^{\circ}$. We need to give the exact answer in terms of $\pi$, and include the correct unit in our answer.

1. Problem Description

The problem asks us to find the area of the shaded region of a circle. The circle has center $O$ and radius of $4$ cm. The central angle of the unshaded region is $90^{\circ}$. We need to give the exact answer in terms of $\pi$, and include the correct unit in our answer.

2. Solution Steps

First, we calculate the area of the entire circle. The formula for the area of a circle is:

$A = \pi r^2$

where $r$ is the radius of the circle. In this case, $r = 4$ cm, so the area of the entire circle is:

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

Next, we find the area of the unshaded sector, which has a central angle of $90^{\circ}$. Since the circle has a total angle of $360^{\circ}$, the unshaded sector is $\frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}$ of the circle. Thus, the area of the unshaded sector is:

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

The shaded area is the difference between the area of the entire circle and the area of the unshaded sector.

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

3. Final Answer

The area of the shaded region is $12\pi \text{ cm}^2$.