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.


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


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.

Try Buzzmath activities for free

and see how the platform can help you.