[Weyl's formula][1] 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).


  [1]: http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum#Weyl.27s_formula