The problem asks for 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 units.

1. Problem Description

The problem asks for 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 units.

2. Solution Steps

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

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

Substituting $r = 4$ cm, we have:

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

Next, we find the area of the sector defined by the central angle $\alpha = 90^{\circ}$. The formula for the area of a sector is:

$A_{sector} = \frac{\alpha}{360^{\circ}} A_{circle}$

Substituting $\alpha = 90^{\circ}$ and $A_{circle} = 16\pi \text{ cm}^2$, we have:

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

3. Final Answer

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