# Platonic Solid

## Platonic Solid

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.