Pentagonal pyramid

Pentagonal pyramid
Type	Johnson J1 - J2 - J3
Faces	5 triangles 1 pentagon
Edges	10
Vertices	6
Vertex configuration	5(32.5) (35)
Schläfli symbol	( ) ∨ {5}
Symmetry group	C5v, [5], (*55)
Rotation group	C5, [5]+, (55)
Dual polyhedron	self
Properties	convex
Net

3D model of a 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 (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 (J₂).

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 as

${\displaystyle (1,0,\tau ),\,(-1,0,\tau ),\,(0,\tau ,1),\,(\tau ,1,0),(\tau ,-1,0),(0,-\tau ,1)}$

where