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 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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* An [[inscribe]]d sphere of radius <math>\frac{s}{2}</math> | * An [[inscribe]]d sphere of radius <math>\frac{s}{2}</math> | ||
* 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 | ||
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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
has:
- Four space diagonals of same lengths
(
) - Surface area of
. (6 sides of areas 
.) - Volume
(
) - A circumscribed sphere of radius
- An inscribed sphere of radius
- A sphere tangent to all of its edges of radius
- A regular tetrahedron can fit in exactly two ways inside a cube
(
)
. (6 sides of areas 
.)
(
)
- For any cube whose circumscribing sphere has radius
, and for any given point in the its 3D dimensional space with distances 
from the cube's eight vertices, we have: ![\[\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.\]](//latex.artofproblemsolving.com/c/6/9/c690a5a8732c460805c5e0bc569e364904e1a785.png)
, and for any given point in the its 3D dimensional space with distances 
from the cube's eight vertices, we have: ![\[\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.\]](http://latex.artofproblemsolving.com/c/6/9/c690a5a8732c460805c5e0bc569e364904e1a785.png)
See also
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