Difference between revisions of "Cube (geometry)"

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A '''cube''', or regular '''hexahedron''', is a solid composed of six [[Square (geometry) | square]] [[face]]s. A cube is [[Platonic solid #Duality | dual]] to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. All edges of cubes are equal to each other.
 
A '''cube''', or regular '''hexahedron''', is a solid composed of six [[Square (geometry) | square]] [[face]]s. A cube is [[Platonic solid #Duality | dual]] to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. All edges of cubes are equal to each other.
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The cube is also a square [[parallelepiped]], an equilateral cuboid, and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
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==Formulas==
 
==Formulas==
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* A sphere [[tangent]] to all of its edges of radius <math>\frac{s\sqrt{2}}{2}</math>
 
* A sphere [[tangent]] to all of its edges of radius <math>\frac{s\sqrt{2}}{2}</math>
 
* A regular tetrahedron can fit in exactly two ways inside a cube
 
* A regular tetrahedron can fit in exactly two ways inside a cube
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* For any cube whose circumscribing sphere has radius <math>R</math>, and for any given point in the its 3D dimensional space with distances <math>d_i</math> from the cube's eight vertices, we have: <cmath>\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.</cmath>
  
 
==See also==
 
==See also==

Latest revision as of 10:04, 24 April 2023

A cube, or regular hexahedron, is a solid composed of six square faces. A cube is dual to the regular octahedron and has octahedral symmetry. A cube is a Platonic solid. All edges of cubes are equal to each other.

The cube is also a square parallelepiped, an equilateral cuboid, and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.


Formulas

A cube with edge-length $s$ has:

  • For any cube whose circumscribing sphere has radius $R$, and for any given point in the its 3D dimensional space with distances $d_i$ from the cube's eight vertices, we have: \[\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.\]

See also

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