A circle is divided by a line, which creates an arc (shown in red). I need to find the radius (r) of the circle, given the following:
- Θ - angle is unknown
- M - arc length is known
- circle center is known
- line parameters are known
I have managed to solve this iteratively, but I'm hoping that there is a more elegant solution.
Let $c$ be the distance from the center of the circle to the line (how you calculate this depends on how exactly your line is expressed). I am assuming that $c<r$. We have $M = r\Theta$ (if $\Theta$ is in radians), and some simple trigonometry shows us that $\cos \frac{\Theta}2 = \frac{c}r$. Putting these two equations together you get $$ \cos\frac{M}{2r} = \frac cr $$ which isn't really solvable for $r$ analytically. But any decent computer or calculator can give you the answer to a reasonable number of decimal places.