Archimedean Solid — Definition, Formula & Examples
An Archimedean solid is a convex polyhedron whose faces are two or more types of regular polygons, arranged so that every vertex looks exactly the same. There are exactly 13 Archimedean solids, including well-known shapes like the truncated icosahedron (soccer ball) and the cuboctahedron.
An Archimedean solid is a convex, isogonal (vertex-transitive) polyhedron composed of two or more types of regular polygons meeting in identical vertex configurations. They are excluded from being Platonic solids (which use only one polygon type) and from being prisms or antiprisms.
How It Works
Each Archimedean solid is identified by its vertex configuration — the sequence of polygon faces meeting at every vertex. For example, the vertex configuration means a triangle and two hexagons meet at each vertex, defining the truncated tetrahedron. You can verify any candidate using Euler's formula: . Many Archimedean solids are created by truncating (cutting off corners of) Platonic solids, which is why several have names like "truncated cube" or "truncated octahedron."
Worked Example
Problem: A truncated octahedron has vertex configuration (4, 6, 6). It has 24 vertices. Use Euler's formula to find the number of edges and faces.
Count faces by type: At each vertex, one square and two hexagons meet. With 24 vertices and each square having 4 vertices, the number of square faces is 24 × 1 / 4 = 6. Each hexagon has 6 vertices, so the number of hexagonal faces is 24 × 2 / 6 = 8. Total faces:
Count edges: Three edges meet at each vertex. Each edge is shared by two vertices, so:
Verify with Euler's formula: Check that V − E + F = 2:
Answer: The truncated octahedron has 36 edges and 14 faces (6 squares and 8 hexagons).
Why It Matters
Archimedean solids appear in chemistry (the truncated icosahedron is the shape of a C₆₀ buckyball molecule), in sports (soccer balls), and in architecture. Studying them deepens understanding of symmetry, vertex configurations, and Euler's formula — all key topics in high-school and college geometry.
Common Mistakes
Mistake: Confusing Archimedean solids with Platonic solids.
Correction: Platonic solids have only one type of regular polygon face (e.g., all equilateral triangles). Archimedean solids use two or more types of regular polygon but still require every vertex to be identical.