Analytical expression for the height of a circular segment

Question

The area of a circular segment is $A=\frac{R^2}{2}\left(\theta - \sin\theta\right)$

Considering $A$ and $R$ known, can you find an analytical expression for $\theta$? Or am I forced to solve it numerically (e.g. Fixed-point iteration)?

Answer 1

Numerical methods are needed.

Given $A$ and $R$, let $f(\theta) = \frac{R^2}{2}\left(\theta - \sin\theta\right) - A$, and hand $f$ to your favorite numerical root-finder.

The function $f$ is monotone increasing, so it only has one root, which is easy to bracket. So your numerical methods don't need to be very sophisticated. You have to be a bit careful if you get very close to $\theta = 0$, but, other than that, I wouldn't expect any problems.