Pentahedron

For the Sylvester pentahedron of a cubic surface, see quaternary cubic.

In geometry, a pentahedron (pl.: pentahedra) is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides and there are two distinct topological types.

With regular polygon faces, the two topological forms are the square pyramid and triangular prism.

Name	Picture	Vertices	Edges	Faces	Faces by type
Square pyramid (Pyramid family)		5	8	5	4 triangles 1 square
Triangular prism (Prism family)		6	9	5	2 triangles 3 squares

The square pyramid can be seen as a triangular prism where one of its side edges (joining two squares) is collapsed into a point, losing one edge and one vertex, and changing two squares into triangles.

Geometric variations with irregular faces can also be constructed.

Some irregular pentahedra with six vertices may be called wedges.

An irregular pentahedron can be a non-convex solid: Consider a non-convex (planar) quadrilateral (such as a dart) as the base of the solid, and any point not in the base plane as the apex.

Hosohedron

There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol {2,5}. It has 2 (antipodal point) vertices, 5 edges, and 5 digonal faces.

External links

• Weisstein, Eric W. "Pentahedron". MathWorld.