polyhedron, closed solid bounded by plane faces; each face of a polyhedron is a polygon. A cube is a polyhedron bounded by six polygons (in this case squares) meeting at right angles. Although regular polygons are possible for any number of sides, there are only five possible regular polyhedrons, having congruent faces, each a regular polygon and meeting at equal angles. The five regular polyhedrons are also known as the Platonic solids, although they were known to the Greeks before the time of Plato. They are the tetrahedron, bounded by four equilateral triangles; the hexahedron, or cube, bounded by six squares; the octahedron, bounded by eight equilateral triangles; the dodecahedron, bounded by twelve regular pentagons; and the icosahedron, bounded by twenty equilateral triangles. The 18th-century Swiss mathematician Leonhard Euler showed that for any simple polyhedron, i.e., a polyhedron containing no holes, the sum of the number of vertices V and the number of faces F is equal to the number of edges E plus 2, or V+F=E+2.

also known as regular polyhedron

Geometric solid all of whose faces are identical regular polygons and all of whose angles are equal. There are only five such polyhedrons. The cube is constructed from the square, the dodecahedron from the regular pentagon, and the tetrahedron, octahedron, and icosahedron (with 20 faces) from the equilateral triangle. They are known as the Platonic solids because of Plato's attempt to relate each to one of the five elements that he believed formed the world.

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