Let $\Omega \subset \mathbb R^n$ a bounded and Lebesgue-measurable set, regular enough to say that its boundary $ \partial\Omega$ is a hypersurface in $ \mathbb R^n$ (for n$\ge 3$). Assuming $\pi_i$ is the projection on the i-th hyperplane, $i \in$ $\{1,. .,n\}$, how can I prove that for every $i \in$ $\{1,. .,n\}$ $$ area(\partial\Omega) \ge 2 area(\pi_i(\Omega)) $$ (I assume area to mean hypersurface-area)?
I tried using the definition of $area(\partial\Omega) = \int (1+|\nabla f|^2)^\frac{1}{2} $, being $f $ a function, regular enough (differentiable with continuous derivatives), such as every point of $ (\partial\Omega)$ is $(x_1, x_2, .., x_{n-1},f(x))$. What can I do now?