Regular Tetrahedron Volume

The regular tetrahedron is the simplest of the five Platonic solids: four vertices, six equal edges, four equilateral faces, and no way to make anything simpler in three dimensions. Its volume from the edge alone is $V = s^3/(6\sqrt{2})$, equivalently $s^3\sqrt{2}/12$, which works out to only about 11.8 percent of the cube built on the same edge. It is a startlingly empty shape for its footprint.

A neat way to see where the constant comes from: take a cube of side $a$ and connect four alternating corners. The result is a regular tetrahedron of edge $a\sqrt{2}$, and the four corner pieces you cut away each have volume $a^3/6$, leaving $a^3/3$ behind. Substituting $s = a\sqrt{2}$ reproduces the formula exactly.

Tetrahedra show up wherever rigidity matters, because a triangle cannot be deformed without changing a side length and a tetrahedron inherits that stubbornness in three dimensions. Space frames, geodesic structures, and the carbon bonds in a diamond lattice all lean on it.