Step-by-step Solution:

The shaded region is the area of the square minus the area of the unshaded, leaf-shaped region common to both quadrants.

Let \(r\) be the radius of the quadrants, which is also the side of the square. Given \(r = 16\) cm.

1. Find the Area of the Unshaded (Leaf-shaped) Region:
The sum of the areas of the two quadrants covers the entire square, but it double-counts the central leaf-shaped region. This can be expressed as:
\(A_{\text{Quadrant 1}} + A_{\text{Quadrant 2}} = A_{\text{Square}} + A_{\text{Leaf}}\).
Rearranging this to solve for the area of the leaf gives:
\[A_{\text{Leaf}} = (A_{\text{Quadrant 1}} + A_{\text{Quadrant 2}}) - A_{\text{Square}}\]
First, calculate the component areas:
\[A_{\text{Square}} = r^2 = 16^2 = 256 \text{ cm}^2\]\[A_{\text{one quadrant}} = \frac{1}{4}\pi r^2 = \frac{1}{4}\pi (16)^2 = 64\pi \text{ cm}^2\]
Now, find the area of the leaf:
\[A_{\text{Leaf}} = (64\pi + 64\pi) - 256 = (128\pi - 256) \text{ cm}^2\]
2. Find the Area of the Shaded Region:
The shaded area is the area of the square minus the area of the leaf.
\[\begin{aligned} A_{\text{Shaded}} &= A_{\text{Square}} - A_{\text{Leaf}} \\ &= 256 - (128\pi - 256) \\ &= 256 - 128\pi + 256 \\ &= 512 - 128\pi \end{aligned}\]
Now, substitute \(\pi \approx \frac{22}{7}\):
\[\begin{aligned} A_{\text{Shaded}} &= 512 - 128 \left(\frac{22}{7}\right) \\ &= 512 - \frac{2816}{7} \\ &= 512 - 402.2857... \\ &\approx 109.7143 \text{ cm}^2 \end{aligned}\]
Result:
The area of the shaded region is approximately \(109.72 \text{ cm}^2\).