Regular Polyhedron — Definition, Formula & Examples
A regular polyhedron is a three-dimensional solid where every face is the same regular polygon and the same number of faces meet at each vertex. There are exactly five regular polyhedra, known as the Platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
A convex polyhedron is regular if and only if all of its faces are congruent regular polygons and its vertex figures are all congruent — that is, the arrangement of faces around every vertex is identical. Equivalently, a regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags (vertex-edge-face triples).
Key Formula
Where:
- = Number of vertices of the polyhedron
- = Number of edges of the polyhedron
- = Number of faces of the polyhedron
Worked Example
Problem: Verify that a regular icosahedron (20 triangular faces, 5 meeting at each vertex) satisfies Euler's formula.
Count faces and edges: Each of the 20 triangular faces has 3 edges, but every edge is shared by 2 faces.
Count vertices: Each vertex has 5 edges meeting it, and each edge connects 2 vertices.
Apply Euler's formula: Substitute into V − E + F.
Answer: The icosahedron has 12 vertices, 30 edges, and 20 faces, and it satisfies Euler's formula: 12 − 30 + 20 = 2.
Why It Matters
Regular polyhedra appear in chemistry (molecular geometry), crystallography, and game design (dice shapes like the d20 are icosahedra). Understanding them builds intuition for symmetry and three-dimensional reasoning needed in courses from geometry through abstract algebra.
Common Mistakes
Mistake: Assuming any solid with regular polygon faces is a regular polyhedron.
Correction: A solid like the cuboctahedron has regular faces (squares and triangles), but they are not all congruent, so it is not a regular polyhedron. Both conditions — identical faces and identical vertex arrangements — must hold.