The Area of a Polygon with an Inscribed Circle

TL;DR

This paper derives formulas for the area of odd-sided polygons with an inscribed circle based on side lengths, extending classical formulas and providing conditions for even-sided cases.

Contribution

It introduces new formulas for the area of odd-sided inscribed polygons based on side lengths and establishes conditions for even-sided polygons.

Findings

Formulas for the area of odd-sided polygons with inscribed circles.

Necessary and sufficient conditions for the existence of solutions for even-sided polygons.

Simplification of the inscribed circle polygon problem compared to cyclic polygons.

Abstract

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 $n$.