Formula: Dodecahedron Volume Side length
$$V ~=~ \frac{15 ~+~ 7 \, \sqrt{5}}{4} \, a^3$$
$$V ~=~ \frac{15 ~+~ 7 \, \sqrt{5}}{4} \, a^3$$
$$a ~=~ \left( \frac{4V}{ 15 ~+~ 7 \, \sqrt{5} } \right)^{1/3}$$
Volume
$$ V $$ Unit $$ \mathrm{m}^3 $$
Volume of a dodecahedron. A regular dodecahedron consists of 12 pentagonal equal faces. The prefactor is approximate:
$$ \frac{15 ~+~ 7 \, \sqrt{5}}{4} ~\approx~ 7.66 $$
Side length
$$ a $$ Unit $$ \mathrm{m} $$
Side length of one side of the pentagon. Since it is a regular polyhedron, all side lengths of a dodecahedron are equal.
For example, with a side length of \( a = 2 \, \text{m} \), the volume of the dodecahedron is: $$ 7.66 \cdot (2 \, \text{m})^3 ~=~ 15.32 \, \text{m}^3 $$