Icosahedron Volume Calculator (Regular)

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):

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:

1. Identical protein subunits can self-assemble in icosahedral symmetry.
2. Icosahedrons enclose the maximum volume for the minimum number of protein subunits.
3. 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. ✓