11. (10 points) As shown in the figure, three line segments divide the regular hexagon into four parts. It is known that the areas of three parts are $2$, $3$, and $4$ square centimeters, respectively. What is the area of the fourth part (the shaded area in the figure) in square centimeters? $\qquad$ square centimeters.
【Solution】Solution: First, let's analyze the regular hexagon. As shown in the left figure, the area of a regular hexagon can be expressed as $6 S$. It is easy to find that the area of $\triangle D E F$ is $S$, and the area of $\triangle C D F$ is $2 S$. As shown in the right figure, connect $D F$ and $C F$. Let the area of the regular hexagon be $6 S$, then the area of $\triangle D E F$ is $S$, and the area of $\triangle C D F$ is $2 S$.
Since $S_{\triangle F D M}: S_{\triangle F C M}=D M: C M=2: 3$,
$$
\begin{array}{l}
\therefore S_{\triangle F D M}=\frac{2}{5} \cdot S_{\triangle F C D}=\frac{4}{5} S, \\
\because S_{F E D M}=S_{\triangle F E D}+S_{\triangle F D M}=S_{F E D N}+S_{\triangle N D M}, \\
\therefore S+\frac{4}{5} S=4+2, \\
\therefore S=\frac{10}{3}, \\
\therefore S_{A B C D E F}=6 S=20, \\
\therefore S_{\text {clear }}=20-2-3-4=11 \mathrm{~cm}^{2} .
\end{array}
$$