A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$.
[img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]
1. **Identify the shaded areas**: The shaded areas are triangles formed by the diagonals of the squares. We need to calculate the area of each triangle separately.

2. **Calculate the area of the first triangle**:
   - The first triangle is formed by the diagonal of the square with side length \(2\) cm.
   - The area of a triangle is given by \(\frac{1}{2} \times \text{base} \times \text{height}\).
   - Here, the base and height are both \(2\) cm.
   \[
   S_1 = \frac{1}{2} \times 2 \times 2 = 2 \text{ cm}^2
   \]

3. **Calculate the area of the second triangle**:
   - The second triangle is formed by the diagonal of the square with side length \(6\) cm.
   - The base and height are both \(6\) cm.
   \[
   S_2 = \frac{1}{2} \times 6 \times 6 = 18 \text{ cm}^2
   \]

4. **Sum the areas of the triangles**:
   - The total shaded area is the sum of the areas of the two triangles.
   \[
   S_{\text{total}} = S_1 + S_2 = 2 + 18 = 20 \text{ cm}^2
   \]

The final answer is \(\boxed{20 \text{ cm}^2}\).