In solving for the distance, d, between two circles of equal radii as a function of that radius, r, such that the area in which they overlap is 1/3 the total area, I have wound up with the following expressions:
$d=2r\cos(\theta/2)$
$\theta-\sin \theta=\pi/2$
Where $\theta$ is the angle between two radii drawn from the center of either circle to the circles' intersection points.
This sort of solves the problem, but since $\theta$ has to be greater than $\pi/2$ the usual approximation $\theta=\sin\theta$ for small $\theta$ is inapplicable, and since sine is a transcendental function, without some sort of approximation like that there's no way to make the analytic solution not-ugly (as far as I can tell).
Is there something I can do here to get a more elegant expression? Could I have gotten a more elegant expression if I did a better job solving the problem at an earlier point?