# Cuboid

Technology
12 hours ago
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2 Albert Flores

### In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cube. In mathematical language a cuboid is a convex polyhedron, whose polyhedral graph is the same as that of a cube.

Special cases are a cube, with 6 squares as faces, a rectangular prism, rectangular cuboid or rectangular box, with 6 rectangles as faces, for both, cube and rectangular prism, adjacent faces meet in a right angle.

## General cuboids

By Euler's formula the numbers of faces F, of vertices V, and of edges E of any convex polyhedron are related by the formula F + V = E + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges. +more

Quadrilaterally-faced hexahedron (cuboid) 6 faces, 12 edges, 8 vertices
Cube (square)Rectangular cuboid (three pairs of rectangles)Trigonal trapezohedron (congruent rhombi)Trigonal trapezohedron (congruent quadrilaterals)Quadrilateral frustum (apex-truncated square pyramid)Parallelepiped (three pairs of parallelograms)Rhombohedron (three pairs of rhombi)
Oh, [4,3], (*432) order 48D2h, [2,2], (*222) order 8D3d, [2+,6], (2*3) order 12D3, [2,3]+, (223) order 6C4v, , (*44) order 8Ci, [2+,2+], (×) order 2

## Rectangular cuboid

Rectangular cuboid
TypePrism Plesiohedron
Faces6 rectangles
Edges12
Vertices8
Symmetry groupD2h, [2,2], (*222), order 8
Schläfli symbol{ } × { } × { }
Coxeter diagram
Dual polyhedronRectangular fusil
Propertiesconvex, zonohedron, isogonal
In a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a right rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. +more

The square cuboid, square box, or right square prism (also ambiguously called square prism) is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol {4} × { }, and its symmetry is doubled from [2,2] to [4,2], order 16.

The cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol {4,3}, and its symmetry is raised from [2,2], to [4,3], order 48.

If the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2(ab + ac + bc).

The length of the space diagonal is : d = \sqrt{a^2+b^2+c^2}.

Cuboid shapes are often used for boxes, cupboards, rooms, buildings, containers, cabinets, books, a sturdy computer chassis, printing devices, electronic calling touchscreen devices, washing and drying machines, etc. Cuboids are among those solids that can tessellate 3-dimensional space. +more

A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example, with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. +more

### Nets

The number of different nets for a simple cube is 11. However, this number increases significantly to 54 for a rectangular cuboid of 3 different lengths.