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Cuboid

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From Wikipedia, the free encyclopedia

Convex polyhedron with six faces with four edges each

For other uses, seeCuboid (disambiguation).

Example of aquadrilateral-facednon-convex hexahedron

Ingeometry, acuboid is ahexahedron withquadrilateral faces, meaning it is apolyhedron with sixfaces; it has eightvertices and twelveedges. Arectangular cuboid (sometimes also called a "cuboid") has allright angles and equal oppositerectangular faces. Etymologically, "cuboid" means "like acube", in the sense of aconvex solid which can be transformed into a cube (by adjusting the lengths of its edges and theangles between its adjacent faces). A cuboid is a convex polyhedron whosepolyhedral graph is the same as that of a cube.^[1]^[2]

General cuboids have many different types. When all of the rectangular cuboid's edges are equal in length, it results in acube, with sixsquare faces and adjacent faces meeting at right angles.^[1]^[3] Along with the rectangular cuboids,parallelepiped is a cuboid with sixparallelogram faces.Rhombohedron is a cuboid with sixrhombus faces. Asquare frustum is a frustum with a square base, but the rest of its faces are quadrilaterals; the square frustum is formed bytruncating theapex of asquare pyramid.In attempting to classify cuboids by their symmetries,Robertson (1983) found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".^[4]

There exist quadrilateral-faced hexahedra which are non-convex.

Some notable cuboids
(quadrilateral-faced convexhexahedra •8 vertices and12 edges each)
Name	Faces	Symmetry group
Cube	6congruent squares	O_h, [4,3], (*432) order48
Trigonal trapezohedron	6 congruentrhombi	D_3d, [2⁺,6], (2*3) order12
Rectangular cuboid	3 pairs ofrectangles	D_2h, [2,2], (*222) order8
Right rhombicprism	1 pair of rhombi, 4 congruentsquares	D_2h, [2,2], (*222) order8
Right squarefrustum	2 non-congruent squares, 4 congruentisosceles trapezoids	C_4v, [4], (*44) order8
Twisted trigonaltrapezohedron	6 congruent quadrilaterals	D₃, [2,3]⁺, (223) order6
Right isosceles-trapezoidal prism	1 pair of isosceles trapezoids; 1,2 or3 (congruent) square(s)	?, ?, ? order4
Rhombohedron	3 pairs of rhombi	C_i, [2⁺,2⁺], (×) order2
Parallelepiped	3 pairs ofparallelograms	C_i, [2⁺,2⁺], (×) order2