Generalization of Heron's formula for $n$-gons
Solution 1:
There cannot a formula for the area of a generic polygon that depends only on the lengths of the sides because polygons are not rigid: they can be deformed by moving the vertices while keeping all side lengths the same.
A formula for the area of cyclic polygons is possible because they are rigid.
Solution 2:
I found this article which provides an interesting generalization for cyclic $n$-gons, so I post it as an answer for future reference.