hexahedron

A hexahedron is a polyhedron with six faces. The name derives from the Greek hex, meaning six, and hedra, meaning face or base, reflecting its six-faced nature.

The cube is the regular hexahedron, the Platonic solid with six congruent square faces. It has 12 edges and 8 vertices. A rectangular prism, or cuboid, is also a hexahedron, with six rectangular faces that need not be squares. Other six-faced solids exist as well; they can be convex or concave and may feature triangles or quadrilaterals as faces. For example, the triangular dipyramid is a convex hexahedron composed of six triangular faces.

In general, a hexahedron has six faces. By Euler’s formula for polyhedra, V − E + F = 2, so with F = 6 one has E = V + 4. The cube specifically satisfies V = 8 and E = 12. Hexahedra appear in various contexts, including geometry and computer-aided design. In computer graphics and engineering, a hexahedral element (often a cube-like cell) is a common six-faced volume used in meshing and finite element analysis, typically with quadrilateral faces.

Overall, the term denotes any six-faced polyhedron, with the cube as the most familiar example. The broader family includes many different shapes, spanning regular, irregular, convex, and non-convex forms, connected by the shared property of having six polygonal faces.