Spherical circle ($C(r)=2\pi R\sin (r/R)$)

On a sphere of radius $R$, a spherical circle of radius $r\in (0,\pi R)$ is the locus at spherical distance $r$ from its center. Its polar angle about that center is $r/R$, so parametrization by longitude has speed $R\sin (r/R)$. Its circumference is therefore $2\pi R\sin (r/R)$. For small $r$, this is $2\pi r-\pi r^3/(3R^2)+O(r^5)$, with the deficit determined by the sphere's positive Gaussian curvature.