View source for Pentahedron

{{Expand Chinese|topic=sci|date=February 2025}}
{{Short description|Polyhedron with five faces}}
{{more sources needed|date=March 2025}}
{{for|the Sylvester pentahedron of a cubic surface|quaternary cubic}}
In [[geometry]], a '''pentahedron''' ({{plural form}}: '''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]].<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}</ref>

<gallery align=center>
Square pyramid.png|[[Square pyramid]]
Triangular prism.png|[[Triangular prism]]
</gallery>

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 [[vertex (geometry)|vertices]] may be called [[wedge (geometry)|wedge]]s.

An irregular pentahedron can be a non-[[Convex polytope|convex]] solid: Consider a non-convex (planar) [[quadrilateral]] (such as a [[dart (geometry)|dart]]) as the base of the solid, and any point not in the base plane as the [[apex (geometry)|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 [[hosohedron|pentagonal hosohedron]] with [[Schläfli symbol]] {2,5}. It has 2 ([[antipodal point]]) vertices, 5 edges, and 5 digonal faces.

==References==
{{Reflist}}

==External links==
*{{MathWorld |urlname=Pentahedron |title=Pentahedron}}

{{Polyhedra}}

[[Category:Polyhedra]]

{{polyhedron-stub}}