Consider a mapping $f:B(x;r) \subset \mathbb{R^2}\longrightarrow \partial \bar{B}(x;r) \subset \mathbb{R}^3$ taking $(y_1,y_2)\mapsto(y_1,y_2, \gamma)$. Then
$$\left|\left|\frac{\partial f}{\partial y_1}\times\frac{\partial f}{\partial y_2}\right|\right|=\left(1+||\nabla\gamma||^2\right)^{1/2}$$
Is there a good intuitive way to understand why this is the scaling factor from a disk to a sphere? How might one know this at a glance, without going through explicit calculations?