= Solution
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 parametrization
$$
\gamma(\theta)=(\sin r\cos\theta,\sin r\sin\theta,\cos r),\qquad0\le\theta\le2\pi.
$$
Its speed is $|\gamma'(\theta)|=\sin r$, so the <arc length> is
$$
\boxed{C(r)=2\pi\sin r.}
$$
For 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.
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