Consider the scalar surface integral:
$$\iint_{S}G(x, y, z) dS=\iint _R\left(G(x, y, f(x,y))\cdot\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\right)dxdy$$
where $S\subset z=f(x,y)$ and $R$ is the projection of $S$ onto the xy plane.
The above formula gives a method for evaluating the surface integral in terms of a standard double integral.
One thing that struck me as interesting, however, was the $\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}$ multiplicand. This seems eerily similar to the arc-length integrand, except one-dimension lower. This similarity is also thus manifested in the surface area of revolution formula.
Is there any deeper connection between why this appears in both?