Example 1 As shown in Figure 1, in hexagon $A B C D E F$, $A B \cong$ $E D$, $A F \cong C D$, $B C \xlongequal[=]{=}$ $F E$, diagonal $F D \perp$ $B D$. It is known that $F D=24 \mathrm{~cm}$, $B D=18 \mathrm{~cm}$. What is the area of hexagon $A B C D E F$ in square centimeters?
Analysis: At first glance, this problem seems intractable, but upon closer inspection, we notice three sets of parallel and equal line segments, as well as two perpendicular diagonals with known lengths. This leads to the idea of translating the shapes to form a rectangle.
Solution: As shown in Figure 2, translate
$\triangle D E F$ to the position of
$\triangle B A G$, and translate
$\triangle B C D$ to the position of
$\triangle G A F$. Then,
the original hexagon is decomposed and recombined
into rectangle $B D F G$. The sides of this rectangle are exactly the known lengths of $B D$ and $F D$. It is easy to see that the area of rectangle $B D F G$ is
$$
24 \times 18=432 \mathrm{~cm}^{2} \text {. }
$$

Therefore, the area of hexagon $A B C D E F$ is $432 \mathrm{~cm}^{2}$.