The problem asks us to find the area of the shaded region of a circle with radius $r = 4$ cm and a central angle $\alpha = 90^\circ$. We need to give the exact answer in terms of $\pi$ and include the correct unit.

1. Problem Description

The problem asks us to find the area of the shaded region of a circle with radius $r = 4$ cm and a central angle $\alpha = 90^\circ$. We need to give the exact answer in terms of $\pi$ and include the correct unit.

2. Solution Steps

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

$A = \pi r^2$

Plugging in the given radius $r = 4$ cm, we get:

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

Next, we find the area of the sector with a central angle of $90^\circ$. Since the central angle is $90^\circ$, the sector is $\frac{90}{360} = \frac{1}{4}$ of the entire circle. The area of the sector is:

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

The shaded region is the sector. Therefore, the area of the shaded region is $4\pi$ cm$^2$.

3. Final Answer

The area of the shaded region is $4\pi$ cm$^2$.