The Area of a Polygon with an Inscribed Circle

Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$K=\sqrt {s(s-a) (s-b) (s-c)}$$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula for polygons of up to eight sides inscribed in a circle. In this paper we derive formulas giving the areas of any $n$-gon, with odd $n$, in terms of the ordered list of side lengths, if the $n$-gon is circumscribed about a circle (instead of being inscribed in a circle). Unlike the cyclic polygon problem, where the order of the sides does not matter, for the inscribed circle problem (our case) it does matter. The solution is much easier than for the cyclic polygon problem, but it does generalize easily to all odd $n$. We also provide necessary and sufficient conditions for there to be solutions in the case of even $n$.