What is a closed form of $P=\prod\limits_{k=3}^{\infty}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}\approx 0.05934871...$ ?
This is the product of the areas of every regular polygon inscribed in a circle of area $1$.
I do not know how to evaluate this product. I have been trying to use complex numbers, to no avail.
Remarks:
$P$ is remarkably close to $\frac{1}{6\pi-2}\approx 0.0593487\color{red}{45}...$ But this is not a closed form, because the product decreases as the number of factors increases, and according to desmos, $$\prod\limits_{k=3}^{10^8}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}<\frac{1}{6\pi-2}$$
There is a related question: What is the product of the circumferences of every regular polygon inscribed in a circle of circumference $1$? This is $P'=\prod\limits_{k=3}^{\infty}\frac{k}{\pi}\sin{\left(\frac{\pi}{k}\right)}\approx 0.51633595...$ The ratio of $P$ to $P'$ is $\prod\limits_{k=3}^{\infty}\cos{\left(\frac{\pi}{k}\right)}\approx 0.11494204...$ which is the Kepler-Boukamp constant and has no known closed form. But is there a closed form of $P$?
EDIT
The product of the areas of every odd-gon inscribed in a circle of area $1$, equals $\frac{\pi}{2}\times$ the Kepler-Boukamp constant, as shown here.