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

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). 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:

$H={\sqrt {\frac {5-{\sqrt {5}}}{10}}}\,a\approx 0.5257\,a.$

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

$A=\left({\frac {\sqrt {25+10{\sqrt {5}}}}{4}}+5{\frac {\sqrt {3}}{4}}\right)a^{2}\approx 3.8855\,a^{2}.$

Its volume when an edge length is known can be figured out with this formula:

$V={\frac {5+{\sqrt {5}}}{24}}\,a^{3}\approx 0.3015\,a^{3}.$

It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J₁₁, one of the 92 Johnson solids 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.

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)