Regular Polyhedrons, definitions and formulas

What is a polyhedron?

A polyhedron is a solid which is limited only by flat surfaces which we call faces, intersections of faces are called edges, and points where edges are cut are called corners. Diagonal lines are called lines that join corners that do not belong to the same face.

What is a regular polyhedron?

As you can imagine, a regular polyhedron is one that all its faces are regular polygons. In total, there are only 5 regular polyhedrons that you already know, each of these polyhedrons has the prefix of the number of faces. Let’s see the figure broken down as it would look without being armed and let’s see the armed figure:

Regular tetrahedron

Regular Hexahedron or cube

Regular octahedron

Regular dodecahedron

Icosahedron regular

Areas and volumes of regular polyhedrons

First we must take into account the following in order to calculate the area, volume and radius of the regular polyhedrons:

A = area
V = volume
a = edge
R = radius of the circumscribed sphere
r = radius of the inscribed sphere
\rho = radius of the sphere tangent to the edges

Formulas for the calculation of the area, volume and radios of a tetrahedron

Area of a tetrahedron

A = a^{2}\sqrt{3} = \cfrac{8}{3} \ R^{2}\sqrt{3} = 24r^{2} \sqrt{3} = 8 \rho \sqrt{3}

Volume of a tetrahedron

V = \cfrac{a^{3}}{12} \sqrt{2} = \cfrac{8}{27} \ R^{3}\sqrt{3} = 8r^{3} \sqrt{3} = \cfrac{8}{3} \ \rho^{3}

Radios of a tetrahedron

R = \cfrac{a}{4} \sqrt{6} , \quad r = \cfrac{a}{12}\sqrt{6}

Formulas for calculating the area, volume and radios of a hexahedron or cube

Area of a cube

A = 6a^{2} = 8R^{2} = 24r^{2} = 12 \rho ^{2}

Volume of a cube

V = a^{3} = \cfrac{8}{9} \ R^{3}\sqrt{3} = 8r^{3} = 2\rho^{3}\sqrt{2}

Radios of a cube

R = \cfrac{a}{2}\sqrt{3}, \quad r = \cfrac{a}{2}

Formulas for the calculation of the area, volume and radios of an octahedron

Area of an octahedron

A = 2a^{3}\sqrt{3} = 4R^{2} \sqrt{3} = 12r^{2}\sqrt{3} = 8 \rho ^{2}\sqrt{3}

Volume of an octahedron

V = \cfrac{a^{3}}{3}\sqrt{2} = \cfrac{4}{3} \ R^{3} = 4 r^{3} \sqrt{3} = \cfrac{8}{3} \ \rho^{3} \sqrt{2}

Radios of an octahedron

R = \cfrac{a}{2}\sqrt{2}, \quad r = \cfrac{a}{6}\sqrt{6}

Formula for the calculation of the area, volume and radios of a dodecahedron

Area of a dodecahedron

A = 3 a^{2}\sqrt{5 \left ( 5 + 2\sqrt{5} \right ) } = 2 R^{2} \sqrt{10 \left ( 5 - \sqrt{5} \right )}

A = 30r^{2} \sqrt{2\left ( 65 - 29 \sqrt{5} \right )} = 6\rho^{2}\sqrt{10 \left (25 - 11 \sqrt{5} \right )}

Volume of a dodecahedron

V = \cfrac{a^{3}}{4} \left ( 15 + 7\sqrt{5} \right ) = \cfrac{2 R^{3}}{9}\left ( 5\sqrt{3} + \sqrt{15} \right )

V = 10r^{3} \sqrt{2 \left ( 65 - 29\sqrt{5} \right )} = 2\rho^{3} \left ( 3\sqrt{5} - 5\right )

Radios of a dodecahedron

R = \cfrac{a}{4}\left ( \sqrt{3} + \sqrt{15} \right ), \quad r = \cfrac{a}{20}\sqrt{10 \left ( 25 + 11 \sqrt{5} \right )}

Formulas for calculating the area, volume and radios of an icosahedron

Area of an icosahedron

A = 5a^{2} \sqrt{3} = 2R^{2}\sqrt{3} \left ( 5 - \sqrt{5} \right )

A = 30r^{2}\sqrt{3} \left ( 7 - 3\sqrt{5} \right ) = 10 \rho^{2}\sqrt{3}\left ( 3 - \sqrt{5} \right )

Volume of an icosahedron

V = \cfrac{5a^{3}}{12} \left ( 3 + \sqrt{5} \right ) = \cfrac{2R^{3}}{3} \left ( \sqrt{10 + 2\sqrt{5}}\right )

V = 10r^{3} \sqrt{3} \left ( 7 - 3 \sqrt{5} \right ) = \cfrac{10 \rho^{3}}{3}\left ( \sqrt{5} - 1 \right )

Radios of an icosahedron

R = \cfrac{a}{4} \sqrt{10 + 2 \sqrt{5}}, \quad r = \cfrac{a\sqrt{3}}{12} \left ( 3 + \sqrt{5} \right )

The formulas mentioned above were taken from the following reference:

Spiegel, M. (1999). Manual de fórmulas y tablas matemáticas, D.F.,México, McGRAW-HILL.

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