How many faces, edges and vertices does a cube have?

Worked example. A cube with 3 cm edges has a surface area of 6 × 3² = 54 cm² and a volume of 3³ = 27 cm³. Double the edge to 6 cm and the area goes up 4 times (216 cm²), the volume 8 times (216 cm³).

The longest straight line inside a cube runs from one corner to the opposite corner. It is twice the circumradius: √3 × a, about 1.732 × a. A 3 cm cube has a corner-to-corner diagonal of about 5.20 cm.

Make one

The net of a cube

A cube has exactly 11 different nets. The cross shape you see first is the famous one, but there are 10 more: strips of four with one square above and one below in different places, staircases, and so on. Press New net above to fold different ones. Shapes like a 2 × 3 block of squares look right but cannot fold into a cube.

You always cut 7 of the 12 edges. Five edges stay as folds and hold the 6 squares together; the other 7 are cut. That is one less than the 8 vertices, the same rule as every solid.

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

Earth, dice and an impossible altar

The word comes from the Greek kybos, “a six-sided die.” It is the only Platonic solid whose name does not end in -hedron.

Plato made it earth. In the Timaeus: “To earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies.”

Doubling the cube. An old story, told by Eratosthenes, says that around 430 BCE the god told the people of Delos to end a plague by building an altar twice the size of the cube-shaped one they had. Doubling the edge makes it 8 times bigger, not 2. The real answer needs an edge of ∛2 times the old one, and in 1837 Pierre Wantzel proved it cannot be drawn with a ruler and compass alone.

Out in the world

Where you see a cube

Salt. Rock salt (halite) grows as cubes. The Handbook of Mineralogy: “Crystals cubic, to 1 m, or octahedral.” Look at table salt under a magnifying glass and you can see tiny cubes.

Filling space. Stack cubes and they fill a room with no gaps. MathWorld: “The cube is the only Platonic solid possessing this property.” That is why boxes, bricks and rooms are built from it.

The Rubik’s Cube. Ernő Rubik, a Hungarian design teacher, “assembled his first cube puzzle in 1974 and called it the Magic Cube,” according to The Strong National Museum of Play. Ideal Toy renamed it the Rubik’s Cube and put it in stores in 1980.

Dice and sacred geometry. The ordinary six-sided die is a cube. Seen corner-on, a cube’s outline is a hexagon with a Y inside, and that exact drawing sits in Metatron’s Cube.

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