Weyl's formula states that
$$N(R)=\frac{V+o(1)}{(4\pi)^{d/2}\Gamma\left(\frac d2+1\right)}R^{d/2},$$ where $d$ is the dimension, $V$ is the volume, and $N(R)$ is number of eigenvalues $\le R$. It works for any compact Riemannian manifold (and many noncompact ones as well).
