Given the circle as seen in the attached image, find the radius of the circle analytically. Is that even possible? I know it can be found numerically. If analytical solution does not exist, can you also provide some hint why not?
If it's not clear: You only know $c$ and $s$.
In the depicted configuration, $c=2R\sin\theta$ and $s=2R(\pi-\theta)$. Let $\varphi=\pi-\theta$ and $k=\frac{c}{s}$.
Given $c$ and $s$, in order to find $\varphi$ we have to solve $\frac{\sin \varphi}{\varphi} = \frac{c}{s}$ or $$ \sin(\varphi) = k\cdot\varphi \tag{1}$$ that has a unique solution $\varphi\in (0,\pi)$ since the sine function is concave over that interval and its derivative is bounded by one in absolute value. $(1)$ has no explicit solution, but Newton's method is very effective in finding an approximate solution. A good starting point for Newton's method can be the solution of the approximate equation: $$ \frac{4}{\pi^2}\varphi(\pi-\varphi) = k\cdot \varphi, \tag{2}$$ i.e. $\displaystyle\varphi_0 = \frac{\pi(4-k\pi)}{4}$.