Truncated Icosahedron — Definition, Formula & Examples
A truncated icosahedron is a 3D solid made by cutting off (truncating) each of the 12 vertices of a regular icosahedron, producing a shape with 12 regular pentagons and 20 regular hexagons. It is the geometric shape behind the classic black-and-white soccer ball.
An Archimedean solid obtained by truncating each vertex of a regular icosahedron at one-third of the edge length, yielding a convex polyhedron with 32 faces (12 regular pentagons and 20 regular hexagons), 90 edges, and 60 vertices, where each vertex is surrounded by one pentagon and two hexagons.
Worked Example
Problem: A truncated icosahedron has 32 faces, 60 vertices, and 90 edges. Verify that it satisfies Euler's formula for polyhedra.
Recall Euler's formula: For any convex polyhedron, the number of vertices minus edges plus faces equals 2.
Substitute the values: Plug in V = 60, E = 90, and F = 32.
Confirm: The left side simplifies to 2, so Euler's formula holds.
Answer: Yes, the truncated icosahedron satisfies Euler's formula: 60 − 90 + 32 = 2.
Why It Matters
The truncated icosahedron appears in chemistry as the structure of the C₆₀ buckyball (buckminsterfullerene), one of the most important molecules in nanotechnology. In geometry courses, it serves as a key example of an Archimedean solid and helps students practice Euler's formula on non-Platonic polyhedra.
Common Mistakes
Mistake: Assuming it is a Platonic solid because it looks highly symmetric.
Correction: Platonic solids have only one type of regular polygon as faces. The truncated icosahedron has two types (pentagons and hexagons), so it is an Archimedean solid, not a Platonic solid.