I have been trying to figure out a way to relate the length of a string that is curved into a circumcircle, along with the distance between the two ends of that string, to the arclength created by that chord. I have drawn a sketch below.
I tried using the following relations (to no avail):
$s=R\theta$
$a=2Rsin(\theta / 2)$
$\theta = 2sin^{-1}\left(\frac{a}{2R}\right)$
$s \approx \sqrt{c^2 + \frac{16}{3}h^2 } $
...all of which involve unknowns. I've even tried combining these equations to find a simplification... again, no avail.
Can this be done?
You have not used all that you know: you know the length of the string, $g$.
Using the original notation from the first diagram for the arc length, the chord and the length of the string and $R$ and $\theta$ as defined in the second diagram:
$$ x = R\theta $$
$$ g = R(2\pi - \theta) $$
$$ e = 2R\sin{\theta \over 2} $$
$g$ and $e$ are known, $x$, $R$ and $\theta$ are unknown. We have three equations in three unknowns, so in theory you could find $x$.
You could for instance take the ratio of the second and third equations to eliminate R and come up with a (not quite straightforward) equation for $\theta$. With $\theta$ known, determining $R$ and $x$ is trivial.