Pentagonal pyramid

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

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

where τ (sometimes written as φ) is the golden ratio.[1]

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:

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

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

${\displaystyle A=\left({\frac {1}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)\approx 3.88554a^{2}.}$

Its volume can be calculated as:

${\displaystyle V=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\approx 0.30150a^{3}.}$[2]

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

Regular pyramids
Digonal	Triangular	Square	Pentagonal	Hexagonal	Heptagonal	Octagonal	Enneagonal	Decagonal...
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

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