My dad asked me this seemingly simple problem:
Suppose you have a (thin) piece of wood in the shape of a circular arc. You can measure the curved length of the wood, $\ell_1$, as well as the shortest distance between the two end points, $\ell_2$.
What is the radius of the original circle, in terms of $\ell_1$ and $\ell_2$?
It feels like it should have a simple solution, but if my reasoning is correct, we have the equations $$\begin{cases}\ell_1 = r\theta\\\ell_2=2r\sin(\frac\theta2),\end{cases}$$ and it doesn't seem trivial to isolate $r$.
If we allow ourselves to make use of the inverse of the $\operatorname{sinc}(x)=\frac{\sin x}x$ function (we can restrict $x$ to $[0,\pi)$ for our purposes), by considering $\ell_2/\ell_1$, we obtain $$r = \frac{\ell_1}{2\operatorname{sinc}^{-1}({\ell_2}/{\ell_1})}$$ which I guess is some sort of an answer. Can we do better? Perhaps some kind of integral representation?
