Let a unit cube in $\mathbb{R}^3$ so that one of its vertices is on the $x-y$ plane, and all other vertices have a positive $z$-coordinate.
I need to show that the height of the cube - namely the highest $z$-coordinate over the points of the cube, is given by the area of its projected area on the $x-y$ plane.
I did get that $z_{\rm max}=|\cos{(\alpha)}|+|\cos{(\beta)}|+|\cos{(\gamma)}|$, which is the projected areas of three non parallel faces of the cube. But what about the other three? is there a way to show that their projection will necessarily fall on the projection of the first three faces?