Other curvatures' π The ratio $\frac{\text{perimeter}}{\text{diameter}}$ of circles on the surface of constant curvature $0$ is constantly $3.14\dots$ and is called $\pi$. 


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*Is this ratio a constant for every other surface of constant curvature $\kappa$?

*How then (by which formula) does this constant depend on  $\kappa$?

On a 2-dimensional sphere of radius $R$, the discs would correspond to the caps on a cone with vertex at the center of the sphere (with opening given by an angle $0<\phi<\pi$). Since the caps are circles of radius $R\sin \phi$ and the diameter would be the length of the arc of the great circle (which has opening $2\phi$), the ratio would be \begin{equation}\frac{c}{d}=\frac{2\pi R\sin \phi}{2R\phi}=\frac{\pi\sin \phi}{\phi}.  \end{equation}
