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 \(\theta_1 , \ldots , \theta_n :\)