# Cube (geometry)

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:

• Four space diagonals of same lengths $s\sqrt{3}$( $\sqrt{s^2+s^2+s^2}=\sqrt{3s^2}=s\sqrt{3}$)
• Surface area of $6s^2$. (6 sides of areas $s \cdot s$.)
• Volume $s^3$( $s \cdot s \cdot s$)
• A circumscribed sphere of radius $\frac{s\sqrt{3}}{2}$
• An inscribed sphere of radius $\frac{s}{2}$
• A sphere tangent to all of its edges of radius $\frac{s\sqrt{2}}{2}$
• A regular tetrahedron can fit in exactly two ways inside a cube
• 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: $$\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.$$