Calculating the height of a circular segment at all points provided only chord and arc lengths
Solution 1:
Using the notation of the figure you have linked to, we have
\begin{equation} R \sin \frac{\theta}{2} = \frac{a}{2} \end{equation}
we can also write
\begin{equation} \theta = \frac{s}{R} = \frac{2 s \sin \theta/2}{a} \end{equation}
From this equation, you can solve for $\theta$.
Once you have solved for $\theta$, you have
\begin{equation} h = R - R \cos(\theta/2) \end{equation}
Since $R = a/(2 \sin \theta/2)$, we have
\begin{equation} h = \frac{a}{2 \sin \theta/2} \left( 1 - \cos\left(\frac{s \sin\theta/2}{a}\right)\right) \end{equation}