Pentagonal pyramid

Short description: 2nd Johnson solid (6 faces)

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

File:J2 pentagonal pyramid.stl

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 apex). 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 Template:Tau (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:

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

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:

Pentagonal frustum is a pentagonal pyramid with its apex truncated

The top of an icosahedron is a pentagonal pyramid

Example

Pentagonal pyramid (at Matemateca IME-USP)

External links

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