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.

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