Regular Tetrahedron Volume

Regular Tetrahedron Volume: V = a³/(6√2)

Of the five Platonic solids, the regular tetrahedron is the leanest one. It has four equilateral triangular faces that are all congruent, four vertices, and six edges, every edge measuring a. Its volume works out to V = a³ / (6√2) ≈ 0.1178·a³. Take an edge of a = 10 and you get V ≈ 117.85 cubic units. Surface area is A = √3·a², while the height running from a vertex down to the face across from it comes to h = a·√(2/3). The shape is its own dual, and it is the one Platonic solid where every face touches every other face. You see it in the methane molecule (CH4), with four hydrogen atoms parked at the vertices around a central carbon, and it underlies the diamond lattice too.

Applications

It turns up in chemistry (sp3 hybridization, the geometry of methane CH4 and ammonia NH3), in packaging (Tetra Pak cartons; the original 1944 design was a regular tetrahedron, which is where the brand name came from), in lightweight geodesic and space-frame structures, in computer graphics (mesh tetrahedralization for finite element analysis), in gaming dice (the d4), and in the crystallography of diamond and silicon.

FAQ

Why does √2 appear in the formula? It traces back to the height. Project a vertex onto the centroid of the triangle opposite it and you pick up a √(2/3) factor; once you take base area times height over three, the algebra tidies up into 1/(6√2).

How does it compare to a cube of the same edge? A cube of edge a has volume a³, so the tetrahedron fills roughly 11.78% of that. Eight regular tetrahedra plus one regular octahedron will tile a cube of edge 2a (a rhombic arrangement, really; tetrahedra on their own can't fill space).

What is the dihedral angle? Between any two adjacent faces it's arccos(1/3) ≈ 70.53°. That's also why four soap bubbles meeting at a point settle into 109.47° tetrahedral angles, which is the supplement of that figure.