What is a tetrahedron?

Worked example. Take a tetrahedron with 6 cm edges. Surface area = √3 × 6² = 36√3 ≈ 62.35 cm². Volume = (√2 ÷ 12) × 6³ = 18√2 ≈ 25.46 cm³. Type 6 into the box above and you get the same numbers.

Why the volume is so small. A tetrahedron is a pyramid, and every pyramid holds one third of base area × height. MathWorld writes it exactly that way: “Since a tetrahedron is a pyramid with a triangular base, V = ⅓ A_bh.” A neat check: the ball that fits inside it (inradius) is exactly one third the size of the ball around it (circumradius).

Make one

The net of a tetrahedron

A regular tetrahedron has exactly 2 different nets. One is a big triangle made of four small ones. The other is a strip of four triangles in a zigzag. Press New net above a few times and you will only ever see those two shapes, turned different ways.

You always cut 3 of the 6 edges. The net has to stay in one piece, so 3 edges stay as folds and 3 get cut. That is one less than the number of vertices, and it works for every solid: cut edges = vertices − 1.

To build it: press Print this net, cut along the solid lines, fold along the dashed lines, put glue on the grey tabs and close it up one face at a time. Albrecht Dürer printed some of the first nets like this in his 1525 book on measuring with compass and ruler. All five Platonic solids, with every net, are in Platonic Solids 3D.

Name and history

Fire, a name and a pyramid

The name is Greek: tetra, four, and hedra, a seat or base. English took it in the 1560s, defined as a “triangular pyramid, solid figure contained by four plane triangular surfaces.”

Plato made it fire. In the Timaeus, around 360 BCE, he picks the sharpest, lightest solid for the element that cuts and burns: “the pyramid is the solid which is the original element and seed of fire.” Earth got the cube, air the octahedron and water the icosahedron.

It is the only Platonic solid that is its own dual. Put a dot in the middle of each of its 4 faces and join the dots: you get another tetrahedron, upside down. Push two tetrahedra through each other and you get the eight-pointed star Kepler named the stella octangula in 1611, better known today as the merkaba.

Out in the world

Where you see a tetrahedron

Methane. A carbon atom with four bonds points them at the four corners of a tetrahedron. Chemistry LibreTexts: “the tetrahedral bond angle of H−C−H is 109.5°.” It calls this “the basic shape of all compounds in which a carbon atom is bonded to four other atoms.”

Milk cartons. Tetra Pak was set up in Lund, Sweden in 1951, and in 1952 it delivered its first filling machine for “tetrahedron-shaped cartons” to the Lund dairy. The company is named after the shape.

Roman caltrops. These iron spikes were thrown on the ground against horses and soldiers. Museum Wales describes them as “four iron spikes joined at the base to project at angles so that, however they lie, three form a tripod and the fourth points upwards.” The four spikes point to the corners of a tetrahedron, which is why one always points up.

Four-sided dice. A fair die needs identical faces, and the tetrahedron is the smallest solid that has them.

The tetrahedron is one of exactly five Platonic solids, and Euclid proved around 300 BCE that there can never be a sixth.