If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using:
$A_n =\overbrace{\pi r^2}^\text{Area of circle}-\overbrace{\frac{1}{2} r^2 n \sin (\frac{2 \pi}{n})}^\text{Area of n-gon} =r^2(\pi-\frac{1}{2} n \sin (\frac{2 \pi}{n}))$
I want to find the following sum (starting with $n=3$, i.e. the $n$-gon is a triangle):
$\Lambda=\sum_{n=3}^{\infty}A_n = r^2\sum_{n=3}^{\infty}(\pi -\frac{1}{2} n \sin (\frac{2 \pi}{n})) = r^2 \lim_{k \rightarrow \infty} (\pi (k-3)-\frac{1}{2} \sum_{n=3}^{k} n \sin (\frac{2 \pi}{n})) $
I have not tested this sum for convergence, but a quick numerical estimate reveals that, if $k=100000, r=1, \Lambda \approx 7.417$. Increasing $k$ gives the same approximation, suggesting convergence.
Does this series converge, and if so, does it have a closed form? Can we find if $\Lambda$ is rational?