The length of an arc within two intersecting circles I found this mathematical expression for the length of a arc $l(r)$ i.e the shorter arc ACB. In other words, why is $l(r)$ equal to that expression with respect to R i.e $l (r) = 2r \arccos (r/2R)$? I have tried hard to prove it but I couldn't. I hope someone could give the clue to the answer. The diagram and question can be found the the attached picture 


We have
$$
l(r) = 2r \angle ADC.
$$
So we just need to compute $\angle ADC$. Let $Q$ be the center of the circle on the left. Note that $AD=r$ and $DQ=AQ=R$. It follows that $\angle ADQ=\angle ADC=\arccos(r/2R)$. To see why, note that $\triangle ADQ$ is isosceles and drop a perpendicular from $Q$ to the midpoint of $AD$.
Is that clear?
