Icosahedron Volume Calculator (Regular)
Compute regular icosahedron volume from edge length.
For d20 dice, virus capsid modeling, and geodesic dome geometry foundations.
A regular icosahedron has 20 congruent equilateral triangle faces, 12 vertices, and 30 edges. It’s the largest Platonic solid by face count.
V = (5 × (3 + √5) / 12) × s³ ≈ 2.1817 × s³
Where s is the edge length.
Worked example — d20 die for tabletop gaming: The iconic 20-sided die (RPG essential). Standard size: s = 16 mm. V = 2.1817 × 4,096 ≈ 8,936 mm³ ≈ 8.94 cm³.
At plastic density 1.2 g/cm³: ~10.7 g per die. About 5× the volume of a d4 with the same edge length.
The d20 is the most recognizable die in tabletop gaming — used for the iconic D&D “to-hit” rolls.
Worked example — geodesic dome foundation: A small geodesic dome (Class I, frequency 1) is based on an icosahedron with 20 triangular panels. For a dome with effective radius of 4 m, the equivalent icosahedron has edge length approximately s = r × √(50 − 10√5) / 5 ≈ 4 × 1.0515 ≈ 4.21 m. V = 2.1817 × 74.6 ≈ 162.7 m³.
This is the geometric volume — actual geodesic domes only use the “upper half” or “5/8 sphere” portion, so the enclosed interior is about 60-80% of this.
Where icosahedra appear in real measurements:
- d20 dice. Tabletop RPG iconic die. The 20-sided shape is roughly spherical and rolls smoothly.
- Geodesic domes (Class I). Buckminster Fuller’s dome designs use icosahedral or octahedral geometries as the starting shape, subdivided for higher frequencies.
- Virus capsids. Many viruses (rhinoviruses, herpesviruses, adenoviruses) have icosahedral protein shells. This is one of the most efficient ways to enclose volume with minimum protein.
- Carbon Buckminsterfullerene (C60). The “buckyball” molecule has truncated icosahedral structure — soccer-ball-like with 12 pentagons and 20 hexagons.
- Some pollen grains and protozoa shells. Microscopic biological structures often have icosahedral symmetry.
- 3D dice for craps-style games. Some specialty casino dice use icosahedral or higher-symmetry shapes.
The golden ratio appears here too:
Like the dodecahedron, the icosahedron has many measurements involving the golden ratio φ = (1 + √5)/2:
- Inradius (insphere): s × φ² / (2√3) ≈ 0.7558 × s
- Circumradius (circumsphere): s × √(φ² + 1) / 2 ≈ 0.9510 × s
- The vertices of an icosahedron lie on three mutually perpendicular golden rectangles.
This is no coincidence — the icosahedron and dodecahedron are “duals” of each other (icosahedron has 20 faces and 12 vertices; dodecahedron has 12 faces and 20 vertices). They share the same symmetry group, and many measurements interrelate.
Useful icosahedron measurements (all derived from s):
| Quantity | Formula | Value for s = 1 |
|---|---|---|
| Edge length | s | 1 |
| Face area (equilateral triangle) | (√3 / 4) × s² | 0.433 |
| Total surface area | 5√3 × s² | 8.660 |
| Volume | (5(3 + √5) / 12) × s³ | 2.182 |
| Vertex-to-vertex (across) | s × φ × √2 (approx) | 2.218 |
| Inradius | s × φ² / 2√3 | 0.756 |
| Circumradius | s × √(φ² + 1) / 2 | 0.951 |
Comparing volumes for the same edge length:
- Tetrahedron: 0.118 × s³
- Cube: 1.000 × s³
- Octahedron: 0.471 × s³
- Dodecahedron: 7.663 × s³
- Icosahedron: 2.182 × s³
For the same edge length, dodecahedra are biggest, then icosahedra, then cubes, octahedra, and tetrahedra. The order roughly matches “how spherical” each shape is.
Why viruses use icosahedral shapes:
Caspar-Klug theory (1962) explains that viruses build icosahedral capsids because:
- Identical protein subunits can self-assemble in icosahedral symmetry.
- Icosahedrons enclose the maximum volume for the minimum number of protein subunits.
- The shape is mechanically stable under stress.
A virus with a 60-protein capsid has exactly 3 proteins per face × 20 faces. Many viruses use 180, 240, or 540 proteins, all multiples of 60 with various subdivisions of the icosahedron.
Sanity check:
- s = 0: V = 0. ✓
- s = 1: V = 5(3 + √5)/12 = (15 + 5√5)/12 ≈ 2.1817. ✓