Icosahedron Calculator

What is a Regular Icosahedron?

A regular icosahedron is a Platonic solid with 20 congruent equilateral-triangle faces, 30 edges of equal length, and 12 vertices where exactly five triangles meet. It is the largest and most complex of the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron), and it most closely approximates a sphere. The golden ratio (φ = (1 + √5)/2) appears throughout its geometry because its 12 vertices can be placed at cyclic coordinates involving zero, one, and φ.

The icosahedron is familiar to many as the standard 20-sided die (d20) used in tabletop role-playing games like Dungeons & Dragons. In nature, many spherical viruses (including poliovirus, herpesvirus, and adenovirus) have icosahedral capsids — protein shells built from repeating subunits arranged with icosahedral symmetry.

Formulas and Calculations

All properties of a regular icosahedron follow from a single dimension — the edge length (a). The golden ratio φ = (1 + √5)/2 ≈ 1.618 appears throughout.

Volume (V): V = (5/12)(3 + √5) × a³ = 5φ²a³/6

The volume is proportional to the cube of the edge length, with the coefficient (5/12)(3 + √5) ≈ 2.1817.

Surface Area (S): S = 5√3 × a²

The icosahedron has 20 equilateral-triangle faces, each with area (√3/4)a², so the total is 20 × (√3/4)a² = 5√3 × a².

Circumradius (R): R = a × √(10 + 2√5) / 4

The circumradius is the distance from the center to any vertex (the radius of the smallest sphere containing the icosahedron).

Midradius (ρ): ρ = aφ/2

The midradius is the distance from the center to the midpoint of any edge.

Inradius (r): r = aφ² / (2√3)

The inradius is the distance from the center to the center of any face (the radius of the largest sphere inside the icosahedron).

How to Use the Icosahedron Calculator

Select what value you know from the dropdown menu: edge length, volume, or surface area. Enter the known value. The calculator will instantly compute all other properties of the regular icosahedron including the edge, volume, surface area, circumradius, midradius, and inradius.