Example 10 The six interior angles of the convex hexagon $A B C D E F$ are all equal, $A B=a, B C=b, C D=c, D E=d$, find the area $S$ of the hexagon $A B C D E F$.
As shown in Figure 1.4.11, by the given conditions, the hexagon can be completed to form an equilateral $\triangle L M N$. Let $E F=x, F A=y$, then
$$
y+a+b=b+c+d=d+x+y
$$

Solving, we get $x=a+b-d, y=c+d-a$. Clearly, $\triangle E N D$, $\triangle B C M, \triangle A F L$ are all equilateral triangles. Therefore,
$$
S=\frac{\sqrt{3}}{4}\left[(b+c+d)^{2}-b^{2}-d^{2}-y^{2}\right]=\frac{\sqrt{3}}{4}\left[2(a+b)(c+d)-a^{2}-d^{2}\right]
$$