Pentagonal pyramid

Summary

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.

Pentagonal pyramid
Pentagonal pyramid.png
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
Pentagonal pyramid flat.svg
3D model of a pentagonal pyramid

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 coordinatesEdit

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

 

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:

 [2]

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

 [3][2]

Its volume can be calculated as:

 [3]

Related polyhedraEdit

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

ExampleEdit

 
Pentagonal pyramid (at Matemateca IME-USP)

ReferencesEdit

  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.

External linksEdit

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