circlesgeometry
Suppose that in the following diagram, I am only given c and s, the lengths of a chord and its segment.
Is it possible to find the exact value of the radius using c and s?
Every formula I found involves using h (distance from centre of chord to centre of the arc) or theta but I am only given c and s. Is there a simple way to find h or theta then use it to find R?
Using the notation of the figure you have linked to, we have
\begin{equation} R \sin \frac{\theta}{2} = \frac{a}{2} \end{equation}
we can also write
\begin{equation} \theta = \frac{s}{R} = \frac{2 s \sin \theta/2}{a} \end{equation}
From this equation, you can solve for $\theta$.
Once you have solved for $\theta$, you have
\begin{equation} h = R - R \cos(\theta/2) \end{equation}
Since $R = a/(2 \sin \theta/2)$, we have
\begin{equation} h = \frac{a}{2 \sin \theta/2} \left( 1 - \cos\left(\frac{s \sin\theta/2}{a}\right)\right) \end{equation}
If you assume the center of the circle to be the origin of a coordinate system, then the coordinates of one of your points (point $B$ in the illustration I added to your question) has coordinates $\left(d, \sqrt{r^2-d^2}\right)$. From that you can deduce the angle $\varphi$ between the horizontal $x$ axis and the line connecting that point to the origin. That angle will satisfy
$$\tan\varphi = \frac{\sqrt{r^2-d^2}}{d}$$
I've marked $\varphi$ in the above illustration. Now you can use that to compute the angle, and from the angle the arc length:
\begin{align*} \varphi &= \arctan\frac{\sqrt{r^2-d^2}}{d} \\ a_{AB} &= 2r\varphi = 2r\arctan\frac{\sqrt{r^2-d^2}}{d} \end{align*}
If you compute this using a pocket calculator, make sure you have its angle measurement mode set to radians, since a result in degrees won't work for the conversion from angle to arc length.
Best Answer
$$S=R\theta$$
$$c=2R\sin\left(\frac{\theta}{2}\right)$$
Dividing we get,
$$\frac{c}{s} = \frac{\sin(t)}{t}$$
where $t=\theta/2$ and so $0< t < \pi/2$. Since $\sin(x)/x$ is 1-1 on $(0,\pi/2)$, a unique solution $t$ exists (assuming $2s/\pi < c < s$) and $R = \frac{s}{2t}$