TITLE: Euclidean Distance on a Sphere
QUESTION [6 upvotes]: I have that the Euclidean distance on the surface of a sphere in terms of the angle they subtend at the centre is
$(\sqrt{2})R\sqrt{1-\cos(\theta_{12})}$
(Where $\theta_{12}$ is the angle that the two points subtend at the centre.)
Why is this; what is the proof?
Cheers, Alex

REPLY [4 votes]: Consider the diagram:
$\hspace{4cm}$
Using the identity $\cos(\theta)=1-2\sin^2(\theta/2)$, the distance is
$$
2r\sin(\theta/2)=r\sqrt{2-2\cos(\theta)}
$$