 Pentagonal Pyramid
Get Pentagonal Pyramid essential facts below. View Videos or join the Pentagonal Pyramid discussion. Add Pentagonal Pyramid to your PopFlock.com topic list for future reference or share this resource on social media.
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 ? (sometimes written as ?) 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 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.