Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 12A / i

For the spherical circle, use a rotation to put the center at the north pole. Points at spherical distance $r$ have polar angle $r$ and admit the parametrizationIts speed is $|{\gamma }^{\prime }(\theta )|=\sin r$, so the arc length isFor small $r$, the Taylor series gives $C(r)=2\pi r-(\pi /3)r^3+O(r^5)$. Thus its circumference is smaller than the Euclidean circumference $2\pi r$ for the same intrinsic radius. Their ratio is $\sin r/r=1-r^2/6+O(r^4)$ and tends to one. The leading agreement expresses local Euclidean behavior; the cubic deficit reflects the positive Gaussian curvature of the unit sphere.