If you know the length of a chord in a circle as well as the length of the circumference minus the segment cut off by the chord, can you find the circumference of the circle?
For example, given the figure below, assume we know $d$, the arc-length of the blue arc, as well as $b$, the length of the red chord. We would like to calculate either the angle $\theta$, the radius, $r$, or the full circumference of the circle.
The following equations can all be derived from the figure:
$\cos\,\theta=\frac{b}{2\,r}$
$d=c\,\frac{\pi-\theta}{\pi}=2\pi\,r\frac{\pi-\theta}{\pi}=2r(\pi-\theta) \quad \textrm{where c is the circumference of the circle}$
so
$r=\frac{d}{2(\pi-\theta)}=\frac{d}{2(\pi - \cos^{-1}\,\frac{b}{2\,r})}$
But I have no idea how to solve that.