Generalization of Heron's formula for $n$-gons

Question

Recently I knew about Heron's formula for the area of some triangle, and its generalizations to quadrilaterals by Bretschneider's formula. According to Wikipedia there are also generalizations for pentagons and hexagons inscribed in a circle.

My question: has someone researched on the possibility to generalize this formula for $n$-gons? Is there any underlying reason for this to be a difficult (or impossible) task?

Thanks!

Answer 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.

Answer 2

I found this article which provides an interesting generalization for cyclic $n$-gons, so I post it as an answer for future reference.