[Math] Formula to find radius of circle, given length of arc, and width of arc

Question

Let's say I have the Arc Length, and Width of that Arc… Is there a formula to find the radius of that circle?

Please See picture:
https://i.stack.imgur.com/WxH8Z.jpg

Arc Length is ACB, Arc Width is AB

Answer 1

There is no closed-form formula.

Let $\theta$ be the half aperture angle. Then the half arc length is

$$a:=r\theta$$

and the half arc "width"

$$w:=r\sin\theta.$$

Taking the ratio, you get a transcendental equation

$$\frac{\sin\theta}\theta=\text{sinc }\theta=\frac wa$$ and $$\theta=\text{sinc}^{-1}\dfrac wa$$ (the inverse cardinal sine function), but this is not considered a usual function.

For the same reason, there is no closed-form formula for $r$, because that would create one for $\theta$.

For small angles, you can use the Taylor development to the fourth or even sixth degree, and solve the biquadratic or triquadratic

$$\frac wa=\color{#0247fe}{\text{sinc }\theta}\approx\color{#32cd32}{1-\frac{\theta^2}6+\frac{\theta^4}{120}}\approx\color{magenta}{1-\frac{\theta^2}6+\frac{\theta^4}{120}-\frac{\theta^6}{5040}}$$ then refine with Newton.