Let $f: \mathbb{S}^2 \to \mathbb{R}$ be a $C^\infty$ function. Assume further that $\nabla f \neq 0$ on $\{f=0\} \neq \emptyset$. I would like to show that $$\mathscr H^1(f^{-1}(0))= \lim_{\varepsilon \to 0} \frac{1}{2\varepsilon}\int_{|f|<\epsilon} \chi(f(x)/\varepsilon)|\nabla f(x)|dS(x)$$ where $dS(x)$ is the natural measure on the sphere and $\chi \in C_c^\infty$ is a plateau/bump function that equals $1$ near $0$ (or any function that is constantly $1$ in a neighborhood of $0$).
For one, I know that $\{f=0\}$ is a finite union of circles (modulo diffeomorphism) because it is a compact manifold of dimension 1, and it would be enough to assume that it admits a single connected component.
Furthermore, I assume that we could invoke the implicit function theorem but I am not sure how.
P.S. If anyone could also provide a reference that involves the above formula, that would be great. I looked into Topics on Analysis in Metric Spaces by Ambrosio and Tilli but I could find none.