Name given to each of the five regular convex polyhedra named after Plato, who linked them to the four elements in his treatise *Timaeus*.

### Formulas

The variable *a* corresponds to the edge length of each solid.

- For a regular tetrahedron:

\(A=\sqrt{3}a^{2}\) and \(V=\frac{\sqrt{2}}{12}a^{3}\)

- For a cube:

\(A=6a^{2}\) and \(V=a^{3}\)

- For a octahedron:

\(A=2\sqrt{3}a^{2}\) and \(V=\frac{\sqrt{2}}{3}a^{3}\).

- For a dodecahedron:

\(A=3\sqrt{5\left ( 5+2\sqrt{5} \right )}a^{2}\) and \(V=\frac{15+7\sqrt{5}}{4}a^{3}\)

- For an icosahedron:

\(A=5\sqrt{3}a^{2}\) and \(V=\frac{5\sqrt{14+6\sqrt{5}}}{12}a^{3}\)

### Examples

The 5 Platonic solids:

Regular tetrahedron |
Cube (regular hexahedron) |
Regular octahedron |

Regular dodecahedron |
Regular Icosahedron |

All the faces of a Platonic solid are congruent regular polygons.