My Trig is rusty and I thought that asking here may help speed along the process as I break out my dusty identity cheat sheet.
I have the following equality that I would like to solve for $c$:
$$0.1666a = a - r^2\left(2 \cos^{-1}\left(\frac{c}{2r}\right) - \sin(2 \cos^{-1}\left(\frac{c}{2r}\right)\right)$$
I'm having trouble boiling this down to one instance of $c$ to begin with.
I don't see how you can solve this explicitly. If you write $\theta = 2\cos^{-1}(c/2r)$, the equation you are faced with solving for $\theta$ is $$0.1666a = a - r^2(\theta - \sin\theta)$$ and that can't be solved explicitly as far as I know.