Definition
Regular polyhedron
A regular polyhedron has congruent regular-polygon faces arranged the same way at every vertex.
What it means
Every face must be the same regular polygon, and every corner must be built in the same way. Only five solids pass both tests: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Remember it
One regular face repeated; one corner pattern repeated.
Example
A cube has six congruent square faces. Exactly three squares meet at every vertex, so the face rule and corner rule both hold. Therefore a cube is a regular polyhedron.
Watch for
A triangular prism is not regular: it has triangular faces and rectangular faces, so its faces are not all congruent.
Common mix-up
Regular faces are necessary but not sufficient; the same face arrangement must occur at every vertex.
Exact meaning
A regular polyhedron is a convex solid whose faces are congruent regular polygons, with the same number and pattern of faces meeting at every vertex. These solids are also called Platonic solids.