Regular Icosahedron — Definition, Formula & Examples

A regular icosahedron is a 3D solid made of 20 identical equilateral triangles, with five triangles meeting at every vertex. It is one of the five Platonic solids and has 12 vertices and 30 edges.

A regular icosahedron is a convex polyhedron composed of 20 congruent equilateral triangular faces, 30 edges of equal length, and 12 vertices, each of which is the meeting point of exactly 5 faces. Its symmetry group is the icosahedral group of order 120.

Key Formula

V = \frac{5}{12}(3 + \sqrt{5})\,a^3

Where:

$V$ = Volume of the regular icosahedron
$a$ = Length of each edge

Worked Example

Problem: Find the volume of a regular icosahedron with edge length 2.

Write the formula: Use the volume formula for a regular icosahedron.

V = \frac{5}{12}(3 + \sqrt{5})\,a^3

Substitute a = 2: Cube the edge length and multiply.

V = \frac{5}{12}(3 + \sqrt{5})(2)^3 = \frac{5}{12}(3 + \sqrt{5})(8)

Evaluate: Since √5 ≈ 2.2361, we get 3 + √5 ≈ 5.2361.

V = \frac{40}{12}(5.2361) \approx 3.3333 \times 5.2361 \approx 17.45

Answer: The volume is approximately 17.45 cubic units.

Why It Matters

The regular icosahedron is the shape behind the classic 20-sided die (d20) used in tabletop games. In science, many viruses have icosahedral protein shells, and geodesic domes (like those designed by Buckminster Fuller) are based on subdivided icosahedra.

Common Mistakes

Mistake: Confusing the icosahedron (20 faces, 12 vertices) with the dodecahedron (12 faces, 20 vertices).

Correction: Notice that the face and vertex counts are swapped — these two Platonic solids are duals of each other. The icosahedron has triangular faces, while the dodecahedron has pentagonal faces.