The cube - diagonal and hexagonal cross-section

The cube hides some elegant properties: the length of its main diagonal and a surprising hexagonal cross-section.

Example — The cube: diagonal and hexagonal cross-section

In a cube with side $\ell$, the main diagonal (between two opposite vertices) measures $\sqrt{3}\,\ell$.

If the cube is cut by a plane passing through $3$ midpoints of non-adjacent edges, the cross-section is a regular hexagon. One checks this by observing that the $6$ distances between consecutive midpoints all equal $\ell\sqrt{2}/2$ and that the sides of the hexagon subtend angles of $60°$.

Links

Topics: Synthetic geometry in space
Concepts: Parallelepiped · Polyhedron
Skills: Calculating · Synthetic geometry