In this page, we get mathematical formulas for calculating the area of a circle segment under a few different conditions, if you already know the radius of the circle.
My question is, how do you do the inverse? If I have a dome-shaped segment that I know is a piece of a perfect circle, and I can measure all of its characteristics, can I use this to find the radius of the circle it came from and how large the central angle was?
The Wikipedia article on the circular segment has the diagram below and the equation $$R=\frac h2+\frac {c^2}{8h}$$ so measure $c$ and $h$ and you are there.
Points $A,B$ and $C$ on the circular arc define a triangle $ABC$, for which the sought circle is a circumscribed circle with radius $R$. Then if we denote $a,b,c$ as the side lengths of $\triangle ABC$, the radius can be found as \begin{align} R&=\frac{abc}{\sqrt{4(ab)^2-(a^2+b^2-c^2)^2}} . \end{align}
Edit
This formula is for the general case, when we just have only three distinct points. If we are free to choose any point on the arc, the middle point is the natural choice. Than $a=b$ and the general formula is simplified to just
\begin{align} R&=\frac{a}{\sqrt{4-(\tfrac{c}a)^2}} . \end{align}
Alternatively, measure $x$ and $y$ in the segment:
$\hspace{5cm}$
The angle: $$\tan \alpha=\frac yx \Rightarrow \alpha =\arctan \frac yx.$$ The radius: $$\sin 2\alpha = \frac xR \Rightarrow R=\frac{x}{\sin 2\alpha} \quad \text{or}\\ x^2+(R-y)^2=R^2 \Rightarrow R=\frac{x^2+y^2}{2y}.$$
