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: a d20 for tabletop gaming The iconic 20-sided die. Standard size: s = 16 mm. V = 2.1817 × 4,096 ≈ 8,936 mm³ ≈ 8.94 cm³.

At plastic density 1.2 g/cm³ that is about 10.7 g per die. Cut a d4 to the same 16 mm edge and it holds only 0.483 cm³, so the d20 is 18.5 times the volume, not the 5 times you might guess from the face count. Surface area scales with the number of faces; volume does not, because the icosahedron is also a far rounder shape than the tetrahedron.

The d20 is the most recognisable die in tabletop gaming, thanks to the D&D “to-hit” roll.

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.

That is the volume of the whole solid. A real dome is only the top of it: a hemispherical dome takes half, and the common “5/8 sphere” cut takes 62.5%, so budget the interior at roughly 50 to 63% of the figure above.

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 is a truncated icosahedron, soccer-ball-like, with 12 pentagons and 20 hexagons.
• Some pollen grains and radiolarian shells. Microscopic biological structures often show icosahedral symmetry.
• Novelty and gaming dice. Casino craps uses precision cubes only, but the d20 turns up in everything from RPGs to the old Magic 8-Ball, whose answer block is a floating icosahedron with 20 printed faces.

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: the icosahedron has 20 faces and 12 vertices, the dodecahedron has 12 faces and 20 vertices. They share a symmetry group, so their measurements keep echoing each other.

Useful icosahedron measurements (all derived from s):

The last row is the one people get wrong. An icosahedron is centrally symmetric, so every vertex has an opposite one, and the distance across is simply twice the circumradius, 1.902 s. It is not φ√2 (2.288) and it is not 2.218; both of those float around online and neither is the width of the solid. If a figure for “across the icosahedron” exceeds 1.902 s, it is measuring something that is not there.

Comparing volumes for the same edge length:

• Tetrahedron: 0.118 × s³
• Octahedron: 0.471 × s³
• Cube: 1.000 × s³
• Icosahedron: 2.182 × s³
• Dodecahedron: 7.663 × s³

So for a fixed edge the dodecahedron is by far the biggest. That is a statement about size, not about roundness, and the two get confused constantly. Measure roundness properly, with sphericity, and the icosahedron wins at 0.939 against the dodecahedron’s 0.910. The dodecahedron is bigger because 12 pentagons of side s simply enclose more room than 20 triangles of side s, not because it is closer to a ball.

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. ✓