# The Japanese Theorem for Nonconvex Polygons - The Total Inradius Function

Author(s):
David Richeson

Using the notation given on the previous page, elementary trigonometry, and the polygonal Carnot's theorem we are able to give an explicit formula for the function $$f: \mathcal{P}_n^c = \mathcal{P}_{R,n}^c \rightarrow {\mathbb R} .$$

Let $$P \in \mathcal{P}_n^c .$$ By the polygonal Carnot's theorem,

$f(P) = r_P = (2-n)R + \sum_{k=1}^n d_k$

where $$d_k$$ is the signed distance from the center of the circle to the $$k$$th side of the polygon. Notice that $$d_k > 0$$ precisely when $$\theta_k < \pi .$$ In particular, as we see in Figure 8, $$d_k = R \cos( \theta_k/2) .$$ When $$\theta_k \geq \pi ,$$ $$d_k = -R \cos (\pi - \theta_k / 2) = R \cos( \theta_k / 2) ,$$ as well.

Figure 8

Thus, we obtain the following explicit expression for the total inradius function in terms of $$\theta_1 , \ldots , \theta_n :$$

$f(P) = f(\theta_1, \ldots , \theta_n ) = R \left( 2 - n + \sum_{k=1}^n \cos \left(\frac{\theta_k}{2} \right) \right) .$

David Richeson, "The Japanese Theorem for Nonconvex Polygons - The Total Inradius Function," Loci (December 2013)