I'm struggling with finding a circle radius $(r)$ of circular segment which has given chord lenght $(s)$ and circular segment area $(A)$.
I'm interested only in solution when segment angle $(\alpha)$ is smaller or equal $\pi$.
If consider this condition, limits are $r_{min} = \frac{s}{2} \implies A=\frac{\pi \cdot r^2}{2}; r_{max} = \infty \implies A = 0$
Between them for each $\alpha$ is only one $r$ and only one $A$ (as shown in equation below)
$$ A={\frac {r^{2}}{2}}\left(\alpha -\sin \alpha \right) $$
There are also other ways how to calculate $A$:
$$ A=r^{2}\arccos \!\left({\frac {r-h}{r}}\right)-(r-h){\sqrt {2hr-h^{2}}} $$ $$ A={\frac {1}{64h^{2}}}\left(\left(s^{2}+4h^{2}\right)^{2}\arccos {\frac {s^{2}-4h^{2}}{s^{2}+4h^{2}}}-4sh\left(s^{2}-4h^{2}\right)\right) $$
(Where $h$ is height of the segment - see fig above)
The last equation seems promising, but I'm unable to solve it for $h$.
Also I'm able to solve it numericaly, but this is not what I need.
Many thanks for your suggestions!
