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

$(1,0,\tau ),\,(-1,0,\tau ),\,(0,\tau ,1),\,(\tau ,1,0),(\tau ,-1,0),(0,-\tau ,1)$

where τ (sometimes written as φ) 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:

$H=\left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)a\approx 0.52573a.$

Its surface area A can be computed as the area of the pentagonal base plus five times the area of one triangle:

$A={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\approx 3.88554\cdot a^{2}.$

Its volume can be calculated as:

$V=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\approx 0.30150a^{3}.$

Related polyhedra

The pentagrammic star pyramid has the same vertex arrangement, but connected onto a pentagram base:

Regular pyramids
Digonal	Triangular	Square	Pentagonal	Hexagonal	Heptagonal	...
Improper	Regular	Equilateral		Isosceles
						...
						...

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

Example

Pentagonal pyramid (at Matemateca IME-USP)

External links

• Weisstein, Eric W., "Pentagonal pyramid" ("Johnson solid") at MathWorld.
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra ( VRML model)