Given a rectangle of height $h$ and area $A$, what is the width $c$ of the chord at the base of a circular segment with the same height and area?
I've made a diagram of the problem:
My progress so far has come from manipulating equations from here. The best equation I have is $(\frac{1}{2}) (\frac{c^2}{8 h}+\frac{h}{2})^2 \left(2 \arccos\left[\frac{\frac{c^2}{8 h}-\frac{h}{2}}{\frac{c^2}{8 h}+\frac{h}{2}}\right]-\sin\left[2 \arccos\left[\frac{\frac{c^2}{8 h}-\frac{h}{2}}{\frac{c^2}{8 h}+\frac{h}{2}}\right]\right]\right)=h w$, which according to Mathematica is not solvable for $c$.