Dodecahedron Volume Calculator – Regular Polyhedron Properties
A regular dodecahedron is one of the five Platonic solids recognised since antiquity. It has 12 regular pentagonal faces, 30 edges of equal length, and 20 vertices. Unlike the other Platonic solids, the dodecahedron is deeply intertwined with the golden ratio φ = (1 + √5) / 2 ≈ 1.618 — the ratio appears in its volume, surface area, and all three characteristic radii. This calculator lets you find the volume, surface area, and all three radii from a single edge length input, with results displayed in multiple unit systems.
Volume Formula
For a regular dodecahedron with edge length a, the volume is:
V = (a³ / 4) × (15 + 7√5)
Since √5 ≈ 2.2361, the constant factor evaluates to (15 + 7√5) / 4 ≈ 7.6631, so the formula simplifies to V ≈ 7.6631 × a³. For example, if a = 5 cm then V ≈ 957.9 cm³. The reverse calculation — finding the edge length from a known volume — uses a = ∛(V / 7.6631).
The formula can be derived by decomposing the dodecahedron into 12 congruent pentagonal pyramids radiating from the centre, computing the volume of each pyramid using its base area and apothem height, and summing.
Surface Area Formula
The total surface area of all twelve regular pentagonal faces is:
A = 3 × √(25 + 10√5) × a²
Each regular pentagonal face has area (a² / 4) × √(25 + 10√5), and there are 12 faces, giving the formula above. For a = 5 cm, the surface area is approximately 516.1 cm².
Characteristic Radii
Like all Platonic solids, a regular dodecahedron has three naturally defined concentric spheres:
- Inradius (r = a × √((25 + 11√5) / 40)) — the radius of the inscribed sphere (insphere). This sphere sits inside the dodecahedron touching the centre of every face. For
a = 5 cm,r ≈ 5.566 cm. - Midradius (ρ = a × φ² / 2) — the radius of the midsphere. This sphere passes through the midpoint of every one of the 30 edges. For
a = 5 cm,ρ ≈ 6.545 cm. - Circumradius (R = a × √3 × φ / 2) — the radius of the circumscribed sphere (circumsphere). This sphere passes through all 20 vertices. For
a = 5 cm,R ≈ 7.011 cm.
The Golden Ratio Connection
The dodecahedron's geometry is inseparable from the golden ratio φ ≈ 1.6180339887. The midradius equals a × φ² / 2 and the circumradius equals a × √3 × φ / 2. The diagonals of each pentagonal face have a ratio of φ to the edge length. This connection made the dodecahedron especially significant to ancient Greek philosophers, who associated it with the cosmos.
Supported Units
The calculator accepts edge lengths in millimetres (mm), centimetres (cm), metres (m), inches (in), and feet (ft). Primary results (volume, surface area, and radii) are shown in the selected unit. The volume is also automatically converted and displayed in cm³, m³, in³, ft³, and litres so you can see all common units at once without manual conversion.
How to Use This Calculator
- Enter the edge length of the regular dodecahedron. Any positive real number is accepted.
- Select the unit from the dropdown — millimetres, centimetres, metres, inches, or feet.
- Results appear instantly, showing volume, surface area, inradius, midradius, and circumradius.
- The Volume in Other Units panel shows the same volume expressed in cm³, m³, in³, ft³, and litres.
- Click the copy icon next to any result to copy that individual value, or use Copy All to copy the complete result set to your clipboard.
- Click Reset to restore the default example (edge = 5 cm).
Applications
Dodecahedral geometry appears in many fields. In crystallography, pyrite crystals commonly grow in dodecahedral forms. In game design, the twelve-sided die (d12) used in tabletop role-playing games is a regular dodecahedron. In architecture and design, the shape is used for decorative elements, lamp shades, and structural joints where twelve-fold symmetry is desired. In chemistry and molecular science, dodecahedrane (C₂₀H₂₀) is a molecule whose carbon skeleton forms a perfect dodecahedron. Volume calculations are useful for estimating the capacity of dodecahedral containers, material requirements for 3D printing, and packing-density problems.
Accuracy and Limitations
All calculations use JavaScript's double-precision floating-point arithmetic (IEEE 754), providing approximately 15 significant digits of precision. Displayed results are rounded to 4 significant figures for readability; the copy-to-clipboard value provides 6 decimal places. The calculator assumes a perfect regular dodecahedron — all 30 edges equal, all 12 faces identical regular pentagons. Irregular, truncated, or rhombic dodecahedra require different formulas not covered here.