Let \((M,g)\) denote a Riemannian manifold, and suppose \(V_d\) is the volume of a unit ball in \(\mathbb{R}^d\). Then the volume of a geodesic ball can be approximated by
\[ \Vol B^d_r(p) = r^d V_d (1-\frac{\scal (p)}{6(d_2)}r^2) + O(r^{d+4}). \]This result follows by applying the approximation formula of the metric tensor in normal coordinates on the volume form.