The five platonic solids are:
- Tetrahedron
- Cube
- Octahedron
- Dodecahedron
- Icosahedron
They all are convex regular polyhedra. These five platonic solids have different formulas.
FORMULA OF TETRAHEDRON
Tetrahedron: A tetrahedron has 4 faces, 4 vertices, 6 edges and 3 concurrent edges at a vertex:
\[\large Volume=\frac{\sqrt{2}}{12}\,a^{3}\]
\[\large Surface\;Area=\sqrt{3}\,a^{3}\]
Cube: The cube is a solid which has has 6 faces, 8 vertices, 12 edges and 3 concurrent edges at a vertex:
\[\large Surface\;Area=4\,a^{2}\]
\[\large Volume=a^{3}\]
\[\large Diagonal=\sqrt{3}\:a\]
Octahedron: A Solid which has 8 faces, 6 vertices, 12 edges and 4 concurrent edges at a vertex.
\[\large Surface\;Area=2\sqrt{3}\:a^{2}\]
\[\large Volume=\frac{\sqrt{2}}{3}\:a^{3}\]
Dodecahedron: A solid which has 12 faces, 20 vertices, 30 edges and 3 concurrent edges at a vertex.
\[\large Surface\;Area=30\times a\times ap\]
\[\large Volume=\frac{1}{4}\left(15+7\sqrt{5}\right)a^{3}\]
Icosahedron: A solid which has 20 faces, 12 vertices, 30 edges and 5 concurrent edges at a vertex.
\[\large Surface\;Area=5\sqrt{3}\:a^{2}\]
\[\large Volume=\frac{5}{12}\left(3+\sqrt{5}\:a^{3}\right)\]
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