What's the arc length of an implicit function?

Consider the divergence theorem on the two-dimensional region $\mathcal R = \{(x,y):f(x,y)\le 0\}$ bounded by the curve $\mathcal C = \partial\mathcal R = \{(x,y):f(x,y)=0\}$, $$\iint_{\mathcal R} \nabla\cdot\mathbf v\,\mathrm dA = \oint_{\mathcal C}\mathbf v\cdot\hat{\mathbf n}\,\mathrm d\ell.$$ If we take $\mathbf v=\hat{\mathbf n}=(\nabla f)/\|\nabla f\|$, we have $\mathbf v\cdot\hat{\mathbf n} = 1$, so $$\iint_{\mathcal R} \nabla\cdot\left(\frac{\nabla f}{\|\nabla f\|}\right)\,\mathrm dA = \oint_{\mathcal C}\mathrm d\ell,$$ which is the arc length of the curve.

I don't know if this formula is useful at all, but it does satisfy your requirements.