Pentagonal pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point. 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.
It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J11
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.
Cartesian coordinates
The pentagonal pyramid can be seen as the "lid" of a regular icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J11. From the Cartesian coordinates of the icosahedron, Cartesian coordinates for a pentagonal pyramid with edge length 2 may be inferred aswhere τ is the golden ratio.
The height H, from the midpoint of the pentagonal face to the apex, of a pentagonal pyramid with edge length a may therefore be computed as:
Its surface area A can be computed as the area of the pentagonal base plus five times the area of one triangle:
Its volume can be calculated as:
Related polyhedra
The pentagrammic star pyramid has the same vertex arrangement, but connected onto a pentagram base:Pentagonal frustum is a pentagonal pyramid with its apex truncated | The top of an icosahedron is a pentagonal pyramid |
Dual polyhedron
The pentagonal pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation.| Dual pentagonal pyramid | Net of dual |