How to find angle of arc given arc length and sagitta? Given the length of an arc and the length of sagitta, can you calculate the angle (radians)?
I struggle to work out all the parameters I need. For instance, to calculate the radius I need the length of the sagitta and the chord length (but I don't have that - to get that I need the radius...)
Do I have too few parameters to get a single answer for this?
 A: Let $s$ be the length of the sagittus, $a$ the length of the arc, $r$ the radius, and $\theta$ the central angle.  We are given $a$ and $s$.  We know $a=r\theta$ and $ s = r-r\cos\left(\frac{\theta}{2}\right)$ so that $$
\frac{a}{s}=\frac{\theta}{1-\cos\left(\frac{\theta}{2}\right)}$$  A quick plot of $f(\theta)= \theta/(1-\cos(\theta/2))$ indicates that $f$ is one-to-one on $(0,\pi)$ so yes, $s$ and $a$ determine the angle.  I don't think there is a simple closed-form formula for $\theta,$ however.  You probably have to compute it numerically.
EDIT 
At the OP's request, I'm adding some comments on how to compute this numerically.  First, I think it's a little more convenient to consider the reciprocal of the expression I had before. $$
\frac{s}{a}=\frac{1-\cos\left(\frac{\theta}{2}\right)}{\theta}$$  Let $x=s/a,$ so that we want to solve $$f(\theta)=x\theta + \cos\left(\frac{\theta}{2}\right)-1=0$$ for $\theta,$ with $x$ given.  The secant method is probably a good way to do this.  You need a couple of starting values for $\theta_0$ and $\theta_1$ and you can use $1$ and $2$, I should think.
Let me know if this doesn't work well for you.       
