0
$\begingroup$

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)?

$\endgroup$
  • 1
    $\begingroup$ Numerically is the only way to go on that one. $\endgroup$ – Gerry Myerson Jul 26 '13 at 8:43
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.