Cube

A cube is a three-dimensional solid with six congruent square faces, eight vertices, and twelve edges. All faces are squares and adjacent faces meet at right angles. It is a regular hexahedron and one of the five Platonic solids.

With edge length a, its volume is a^3 and its surface area is 6a^2. The space diagonal is a√3, and each face diagonal is a√2. The distance from the center to a face is a/2, and the distance from the center to a vertex is a√3/2.

In geometry, the cube has full octahedral symmetry (Oh) with 48 symmetries; the rotational symmetries alone number 24. Its standard orientation can be represented by vertices at (±a/2, ±a/2, ±a/2).

Common uses include dice and the Rubik’s Cube, as well as various architectural and packaging applications where a cube’s proportions and volume efficiency are advantageous.

In mathematics, cube also refers to the operation of cubing a number, producing the value n^3 for an input n. The term reflects the object’s three-dimensional nature and its role in volume and cubic relationships.