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?

## Best Answer

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.

EDITAt 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.