Pentahedron: Polyhedron with Five Faces

Two primary convex forms

• Square pyramid: one quadrilateral base and four triangular faces meeting at a single apex. This shape is a common example used in modeling and architecture.
• Triangular prism: two parallel triangular faces joined by three rectangular or parallelogram lateral faces. It is a simple prism with a triangular cross-section.

These two are the only topologically distinct convex pentahedra. If all faces are regular polygons there are just these two familiar shapes. More generally, a pentahedron may be irregular, with nonplanar arrangements of faces or with some faces nonconvex.

Basic properties and construction

Euler's formula for convex polyhedra, V − E + F = 2, constrains the possible numbers of vertices (V) and edges (E) when F = 5: one finds E = V + 3. Counting face-edge incidences and face polygons gives the combinatorial possibilities that lead to the two convex types listed above. Faces meet along edges and at vertices; symmetry varies from the high symmetry of regular pyramids to the lower symmetry of oblique prisms.

Pentahedra can be built from nets, by joining polygonal faces along matching edges. They are simple solids for teaching spatial reasoning and are used in modeling, 3D printing, and as components in composite structures. In crystallography or chemistry the term is less common—molecular shapes are described by bonding geometry rather than face counts.

Notable facts and duals

The dual of any polyhedron swaps faces and vertices: a pentahedron has a dual with five vertices. For example, the dual of the triangular prism corresponds to a triangular bipyramid. For further general background on polyhedra and polygons see entries on polyhedron and regular polygon.

While five-faced solids are simple in count, they illustrate core ideas of topology and combinatorics in three dimensions and serve as useful pedagogical and practical forms in design and engineering.