Euclidean Distance on a Sphere 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

Consider the diagram:

$\hspace{4cm}$!enter image description here

Using the identity $\cos(\theta)=1-2\sin^2(\theta/2)$, the distance is $$ 2r\sin(\theta/2)=r\sqrt{2-2\cos(\theta)} $$