Great Rhombicosidodecahedron — Definition, Formula & Examples
The great rhombicosidodecahedron is an Archimedean solid with 62 faces, 180 edges, and 120 vertices. Its faces consist of 30 squares, 20 regular hexagons, and 12 regular decagons.
The great rhombicosidodecahedron (also called the truncated icosidodecahedron) is the convex uniform polyhedron obtained by truncating an icosidodecahedron. It has vertex configuration , meaning that at every vertex, a square, a regular hexagon, and a regular decagon meet in that cyclic order.
Worked Example
Problem: Verify that the great rhombicosidodecahedron satisfies Euler's formula for polyhedra, given that it has 62 faces, 180 edges, and 120 vertices.
Recall Euler's formula: For any convex polyhedron, Euler's formula states:
Substitute the values: Plug in , , and :
Confirm: The left side equals 2, which matches the right side, so Euler's formula holds.
Answer: . Euler's formula is satisfied.
Why It Matters
The great rhombicosidodecahedron is the largest Archimedean solid by number of faces and vertices. Studying it deepens understanding of symmetry groups and uniform polyhedra, which appears in advanced geometry, crystallography, and architectural design.
Common Mistakes
Mistake: Confusing the great rhombicosidodecahedron (62 faces) with the small rhombicosidodecahedron (62 faces but different face types: 20 triangles, 30 squares, and 12 pentagons).
Correction: Check the face types. The great version has squares, hexagons, and decagons, while the small version has triangles, squares, and pentagons.