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A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.

There are seven topologically distinct convex hexahedra,[1] one of which exists in two mirror image forms. There are three topologically distinct concave hexahedra. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.

Convex, Cuboid[edit]

Quadrilaterally-faced hexahedron (Cuboid) 6 faces, 12 edges, 8 vertices
Hexahedron.png Cuboid no label.svg Trigonal trapezohedron.png Trigonal trapezohedron gyro-side.png Usech kvadrat piramid.png Parallelepiped 2013-11-29.svg Rhombohedron.svg
Rectangular cuboid
(three pairs of
Trigonal trapezohedron
(congruent rhombi)
Trigonal trapezohedron
(congruent quadrilaterals)
Quadrilateral frustum
square pyramid)
(three pairs of
(three pairs of
Oh, [4,3], (*432)
order 48
D2h, [2,2], (*222)
order 8
D3d, [2+,6], (2*3)
order 12
D3, [2,3]+, (223)
order 6
C4v, [4], (*44)
order 8
Ci, [2+,2+], (×)
order 2

Convex, Others[edit]

Hexahedron5.svg Hexahedron7.svgHexahedron7a.svg Hexahedron6.svg Hexahedron2.svg Hexahedron3.svg Hexahedron4.svg
Triangular bipyramid Tetragonal antiwedge. Chiral – exists in "left-handed" and "right-handed" mirror image forms. Pentagonal pyramid
36 Faces
9 E, 5 V Faces
10 E, 6 V Faces
11 E, 7 V
5.35 Faces
10 E, 6 V Faces
11 E, 7 V Faces
12 E, 8 V


There are three further topologically distinct hexahedra that can only be realised as concave figures:

Hexahedron8.svg Hexahedron10.svg Hexahedron9.svg Faces
10 E, 6 V Faces
11 E, 7 V Faces
12 E, 8 V

A digonal antiprism can be considered a degenerate form of hexahedron, having two opposing digonal faces and four triangular faces. However, digons are usually disregarded in the definition of non-spherical polyhedra, and this case is often simply considered a tetrahedron and the four remaining triangular faces considered to compose the full solid.

See also[edit]


External links[edit]