Calculating radius given the arc length and distance

Question

I wanted a formula that can tell the length of the radius of the circle given the arc length and the shortest distance between them.

I have derived it to this form:

$d^2 = 2r^2(1-\cos{\frac{l}{r}})$

d: shortest distance

r: radius of the circle

l: arc length

I understand that it cannot be solved for r algebraically with precise value.

Can someone give me a neat and a very good approximation equation for r

Answer 1

A simpler relation between $l$, $d$, and $r$ is $d = 2r\cdot\sin \frac{l}{2r}$. This is equivalent to yours by applying the half angle or double angle formula.

Note that for small angles, $\sin{x}\approx x$, which gives $d=l$ and $r=\infty$. This is obviously not good enough for your purposes.

Using the approximation $sin{x}\approx x-\frac{x^3}{3!}$, the first two terms of the power series expansion, you do get something useful, namely something that simplifies to: $$r = \frac{L}{\sqrt{ 24(1-\frac{d}{L}) }}$$

That should be reasonably accurate until the arc spans about 60 or 70 degrees.