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:
[latex]A=\sqrt{3}a^{2}[/latex] and [latex]V=\frac{\sqrt{2}}{12}a^{3}[/latex]
- For a cube:
[latex]A=6a^{2}[/latex] and [latex]V=a^{3}[/latex]
- For a octahedron:
[latex]A=2\sqrt{3}a^{2}[/latex] and [latex]V=\frac{\sqrt{2}}{3}a^{3}[/latex].
- For a dodecahedron:
[latex]A=3\sqrt{5\left ( 5+2\sqrt{5} \right )}a^{2}[/latex] and [latex]V=\frac{15+7\sqrt{5}}{4}a^{3}[/latex]
- For an icosahedron:
[latex]A=5\sqrt{3}a^{2}[/latex] and [latex]V=\frac{5\sqrt{14+6\sqrt{5}}}{12}a^{3}[/latex]
Examples
The 5 Platonic solids: |
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| Regular tetrahedron | Cube (regular hexahedron) | Regular octahedron |
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| Regular dodecahedron | Regular Icosahedron |