Pentagonal pyramid

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Pentagonal pyramid
Pentagonal pyramid.png
TypeJohnson
J1 - J2 - J3
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

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[edit]

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 polyhedra[edit]

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

Pentagram pyramid.png
Regular pyramids
Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
Improper Regular Equilateral Isosceles
Biangular pyramid1.png Tetrahedron.svg Square pyramid.png Pentagonal pyramid.png Hexagonal pyramid.png Heptagonal pyramid1.png Octagonal pyramid1.png Enneagonal pyramid1.png Decagonal pyramid1.png
Spherical digonal pyramid.png Spherical trigonal pyramid.png Spherical square pyramid.png Spherical pentagonal pyramid.png Spherical hexagonal pyramid.png Spherical heptagonal pyramid.png Spherical octagonal pyramid.png Spherical enneagonal pyramid.png Spherical decagonal pyramid.png
Pentagonal frustum.svg
Pentagonal frustum is a pentagonal pyramid with its apex truncated
Icosahedron.png
The top of an icosahedron is a pentagonal pyramid

Dual polyhedron[edit]

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
Dual pentagonal pyramid.png Dual pentagonal pyramid net.png

Example[edit]

Pentagonal pyramid (at Matemateca IME-USP)

References[edit]

  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 links[edit]