Example 4 As shown in Figure 6, point $A$ is on the line segment $B G$, quadrilaterals $A B C D$ and $D E F G$ are both squares, with areas of $7 \mathrm{~cm}^{2}$ and $11 \mathrm{~cm}^{2}$, respectively. Find the area of $\triangle C D E$.
Analysis: Since $CD = \sqrt{7}$, we only need to find the height $EH$ on side $CD$ (as shown in Figure 6).

From the given conditions, it is easy to see that $\triangle EDH$ is obtained by rotating $\triangle DAG$ around point $D$ by $90^{\circ}$, so,
$$
EH = AG = \sqrt{DG^2 - AD^2} = 2.
$$

Therefore, $S_{\triangle ECD} = \sqrt{7} \, \text{cm}^2$.