How do you find the volume of a regular dodecahedron from its edge length?

For a regular dodecahedron with edge length a, the volume is given by V = ((15 + 7√5) / 4) · a³. Simply measure the edge length, cube it, and multiply by the constant (15 + 7√5) / 4.

Which units can this dodecahedron volume calculator use?

The calculator supports meters, centimeters, millimeters, inches, feet, and yards. It converts internally so you can freely switch units while keeping the geometry consistent.

Can I recover the edge length from the volume of a regular dodecahedron?

Yes. If V is the volume, the edge length a is obtained from the inverse formula a = cube_root(4V / (15 + 7√5)). The calculator implements this mode so you can input a volume and get the corresponding edge length.

Is the dodecahedron volume formula exact or an approximation?

The formula V = ((15 + 7√5) / 4) · a³ is exact. In practice, numerical results are rounded to a reasonable number of decimal places, but the underlying computation uses the exact constant and double-precision arithmetic.

Regular Dodecahedron Volume Calculator

Compute the volume of a regular dodecahedron from its edge length – or recover the edge length from a known volume. The tool uses the exact formula \( V = \frac{15 + 7\sqrt{5}}{4} a^3 \) and supports both metric and imperial units.

Formula for the volume of a regular dodecahedron

A regular dodecahedron is a Platonic solid with 12 congruent regular pentagonal faces, 20 vertices, and 30 edges. If we denote the edge length by \( a \), its volume is

\( V = \dfrac{15 + 7\sqrt{5}}{4}\, a^{3} \)

The constant \( \dfrac{15 + 7\sqrt{5}}{4} \approx 7.6631 \) is dimensionless. This means that if \( a \) is measured in meters, you obtain the volume in cubic meters; if \( a \) is in centimeters, you get cubic centimeters, and so on.

Step-by-step example

Suppose the edge length is \( a = 2\ \text{cm} \). Then:

Cube the edge length: \( a^{3} = 2^{3} = 8\ \text{cm}^{3} \).
Compute the constant: \( \dfrac{15 + 7\sqrt{5}}{4} \approx 7.6631 \).
Multiply: \( V \approx 7.6631 \times 8 \approx 61.3\ \text{cm}^{3} \).

The calculator performs exactly these steps internally (using the full precision of \( \sqrt{5} \)), then rounds the result to a user-friendly number of decimal places.

Inverse formula: edge length from volume

When the volume \( V \) is known and you want the edge length \( a \), you can invert the formula:

\( a = \sqrt[3]{ \dfrac{4V}{15 + 7\sqrt{5}} } \)

This is the formula used by the volume → edge length mode. You can specify the volume in any supported cubic unit (m³, cm³, mm³, in³, ft³, or yd³); the tool converts internally and returns an edge length in the corresponding linear unit.

How to use the regular dodecahedron volume calculator

Choose the calculation mode. Use “Given edge length → volume” if you know the edge, or “Given volume → edge length” if you know the volume.
Enter the known value. Type the edge length or volume as a positive number, then select the appropriate unit.
Click “Calculate”. The primary result box shows the main quantity (volume or edge length), while the secondary lines show converted values (cubic meters and liters for volume mode, meters for edge length mode).
Adjust units as needed. You can switch units and recompute instantly to compare metric and imperial measurements.

FAQ – Regular dodecahedron volume

What is the volume formula for a regular dodecahedron?

For a regular dodecahedron with edge length \( a \), the volume is \( V = \dfrac{15 + 7\sqrt{5}}{4} a^{3} \). This formula is exact and follows from the geometry of the dodecahedron as a Platonic solid.

Which units can I use for the edge length?

The calculator accepts meters, centimeters, millimeters, inches, feet, and yards. Internally, all values are converted and processed using consistent units to avoid rounding errors.

What is the typical use of a dodecahedron volume calculator?

Typical applications include mathematical exercises in solid geometry, visualization of Platonic solids, 3D printing and modelling of gaming dice or decorative objects, and analytic checks in computational geometry.

Does this calculator work for non-regular dodecahedra?

No. The formula used here is valid only for a regular dodecahedron (all edges equal, all faces regular pentagons). For an irregular dodecahedron with varying edge lengths or face shapes, the volume must be computed using more general 3D geometry methods.