Icosidodecahedron — Definition, Formula & Examples
An icosidodecahedron is a polyhedron with 32 faces — 20 equilateral triangles and 12 regular pentagons — where every edge borders one triangle and one pentagon. It can be thought of as a combination of an icosahedron and a dodecahedron.
The icosidodecahedron is an Archimedean solid with 32 faces (20 triangular, 12 pentagonal), 60 edges, and 30 vertices. At each vertex, two triangles and two pentagons meet in alternating fashion, giving a vertex configuration of (3, 5, 3, 5). It can be constructed by truncating either an icosahedron or a dodecahedron at the midpoints of their edges (a process called rectification).
Worked Example
Problem: An icosidodecahedron has 30 vertices and 32 faces (20 triangles and 12 pentagons). Verify the number of edges using Euler's formula for polyhedra.
Recall Euler's formula: For any convex polyhedron, vertices minus edges plus faces equals 2.
Substitute known values: Plug in V = 30 and F = 32.
Solve for E: Combine the constants and isolate E.
Answer: The icosidodecahedron has 60 edges, which is consistent with Euler's formula.
Why It Matters
The icosidodecahedron appears in architecture, molecular chemistry, and art (such as M.C. Escher's work). Studying it helps you practice applying Euler's formula and distinguishing between Platonic solids and Archimedean solids in geometry courses.
Common Mistakes
Mistake: Confusing the icosidodecahedron with a Platonic solid.
Correction: Platonic solids have only one type of regular polygon as faces. The icosidodecahedron uses two types (triangles and pentagons), making it an Archimedean solid instead.