# Pentagonal pyramid

Pentagonal pyramid
TypeJohnson
J1J2J3
Faces5 triangles
1 pentagon
Edges10
Vertices6
Vertex configuration5(32.5)
(35)
Schläfli symbol( ) ∨ {5}
Symmetry groupC5v, [5], (*55)
Rotation groupC5, [5]+, (55)
Dual polyhedronself
Propertiesconvex
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 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

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

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

${\displaystyle H=\left({\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)a\approx 0.52573a.}$[2]

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

${\displaystyle A={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\approx 3.88554\cdot a^{2}.}$[3][2]

Its volume can be calculated as:

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

## Example

Pentagonal pyramid (at Matemateca IME-USP)

## References

1. ^ Weisstein, Eric W. "Icosahedral Group". mathworld.wolfram.com. Retrieved 2020-04-12.
2. ^ a b Sapiña, R. "Area and volume of a pentagonal pyramid and Johnson solid J₂". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-06-29.
3. ^ a b Weisstein, Eric W. "Pentagonal Pyramid". mathworld.wolfram.com. Retrieved 2020-04-12.