Let $ n \in \mathbb S_2$ be a unit vector, s.t. $n_i > 0$ for i = 1, 2, 3.
Let V be a triangular pyramid with faces $ S_j = \partial V \cap \{ z \in \mathbb R | z_j = 0\}$ and $ S= \partial V \setminus \cup^3_{i=1} S_j $ where $n$ is the normal vector to the face $S$.
Why does $ | S_j | = n_j |S|$ does hold true?
Faces $S_j$ and $S$ share a common edge. The altitude $h_j$ of $S_j$ with respect to that edge is the projection of the altitude $h$ of $S$ onto the plane of $S_j$. But $n$ is perpendicular to $S$ and $n_j$ is the projection of $n$ on a line perpendicular to $S_j$, hence by similitude: $n_j/n=h_j/h=|S_j|/|S|$.