# Pentagonal pyramid

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Pentagonal pyramid | |
---|---|

Type |
Johnson J _{1} - J - J_{2}_{3} |

Faces |
5 triangles 1 pentagon |

Edges | 10 |

Vertices | 6 |

Vertex configuration |
5(3^{2}.5)(3 ^{5}) |

Symmetry group |
C_{5v} |

Dual polyhedron | self |

Properties | convex |

Net | |

In geometry, a **pentagonal pyramid** is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex). Like any pyramid, it is self-dual.

The *regular* pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (*J*_{2}).

Its height *H*, from the midpoint of the pentagonal face to the apex, (as a function of *a*, where *a* is the side length), can be computed as:

, while its surface area, *A*, can be computed as:

It can be seen as the "*lid*" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, *J*_{11}. The 92 Johnson solids were named and described by Norman Johnson in 1966.

More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height.

The volume of a pentagonal pyramid is:

## External links

- Template:Mathworld2
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra ( VRML model)

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