Let $\mu(x)dx$ be a measure in $\mathbb{R}^{2n-2}$, where $\mu$ (a $C^\infty$ and positive function) is the density of the volume in the sense that $\DeclareMathOperator{\Vol}{\mathrm{Vol}} \Vol_\mu(B_r)=\int_{B_r}\mu(x)dx$, where $B_r$ be the ball of radius $r$ on $\mathbb{R}^{2n-2}$. Now suppose that $\mu(x)dx=d\Gamma$: by Stokes's theorem we get $$\Vol_\mu(B_r)=\int_{B_r}\mu(x)dx=\int_{B_r}d\Gamma=\int_{S_r}\Gamma$$ where $S_t$ is the sphere in $\mathbb{R}^{2n-3}$. The question is: can we express the area of $S_r$ as a function of the density $\mu$?