Exercise: Calculate the perimeter of the unit ball in $\mathbb R^n$, i.e. show that $\mathcal H^{n-1}(S^{n-1})=n\omega_n$, where $\mathcal H^{n-1}$ is the Hausdorff measure of dimension $n-1$ and $\omega_n=\mathcal H^{n}(B^n)$.
I think I should use the Gauss-Green Theorem: $\int_E \nabla \phi(x)dx = \int_{\partial_E} \phi\nu_Ed\mathcal H^{n-1}$ for every $\phi \in C^1_c(\mathbb R^n)$, where $\nu_E$ is outer normal.