Pentagonal pyramid

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Pentagonal pyramid
Pentagonal pyramid.png
Type Pyramid
Johnson
J1J2J3
Faces 5 triangles
1 pentagon
Edges 10
Vertices 6
Vertex configuration [1]
Symmetry group
Dihedral angle (degrees)As a Johnson solid:
  • triangle-to-triangle: 138.19°
  • triangle-to-pentagon: 37.37°
Dual polyhedron self-dual
Properties convex,
elementary (Johnson solid)
Net
Pentagonal pyramid flat.svg

In geometry, pentagonal pyramid is a pyramid with a pentagon base and five triangular faces, having a total of six faces. It is categorized as a Johnson solid if all of the edges are equal in length, forming equilateral triangular faces and a regular pentagonal base. The pentagonal pyramid can be found in many polyhedrons, including their construction. It also occurs in stereochemistry in pentagonal pyramidal molecular geometry.

Contents

Properties

A pentagonal pyramid has six vertices, ten edges, and six faces. One of its faces is pentagon, a base of the pyramid; five others are triangles. [2] Five of the edges make up the pentagon by connecting its five vertices, and the other five edges are known as the lateral edges of the pyramid, meeting at the sixth vertex called the apex. [3] A pentagonal pyramid is said to be regular if its base is circumscribed in a circle that forms a regular pentagon, and it is said to be right if its altitude is erected perpendicularly to the base's center. [4]

Like other right pyramids with a regular polygon as a base, this pyramid has pyramidal symmetry of cyclic group : the pyramid is left invariant by rotations of one, two, three, and four in five of a full turn around its axis of symmetry, the line connecting the apex to the center of the base. It is also mirror symmetric relative to any perpendicular plane passing through a bisector of the base. [1] It can be represented as the wheel graph ; more generally, a wheel graph is the representation of the skeleton of a -sided pyramid. [5] It is self-dual, meaning its dual polyhedron is the pentagonal pyramid itself. [6]

3D model of a pentagonal pyramid J2 pentagonal pyramid.stl
3D model of a pentagonal pyramid

When all edges are equal in length, the five triangular faces are equilateral and the base is a regular pentagon. Because this pyramid remains convex and all of its faces are regular polygons, it is classified as the second Johnson solid . [7] The dihedral angle between two adjacent triangular faces is approximately 138.19° and that between the triangular face and the base is 37.37°. [1] It is elementary polyhedra, meaning it cannot be separated by a plane to create two small convex polyhedrons with regular faces. [8] Given that is the length of all edges of the pentagonal pyramid. A polyhedron's surface area is the sum of the areas of its faces. Therefore, the surface area of a pentagonal pyramid is the sum of the four triangles and one pentagon area. The volume of every pyramid equals one-third of the area of its base multiplied by its height. That is, the volume of a pentagonal pyramid is one-third of the product of the height and a pentagonal pyramid's area. [9] In the case of Johnson solid with edge length , its surface area and volume are: [10]

Applications

In polyhedron

Pentagonal pyramids can be found in a small stellated dodecahedron SmallStellatedDodecahedron.gif
Pentagonal pyramids can be found in a small stellated dodecahedron

Pentagonal pyramids can be found as components of many polyhedrons. Attaching its base to the pentagonal face of another polyhedron is an example of the construction process known as augmentation, and attaching it to prisms or antiprisms is known as elongation or gyroelongation, respectively. [11] Examples of polyhedrons are the pentakis dodecahedron is constructed from the dodecahedron by attaching the base of pentagonal pyramids onto each pentagonal face, small stellated dodecahedron is constructed from a regular dodecahedron stellated by pentagonal pyramids, and regular icosahedron constructed from a pentagonal antiprism by attaching two pentagonal pyramids onto its pentagonal bases. [12]

Some Johnson solids are constructed by either augmenting pentagonal pyramids or augmenting other shapes with pentagonal pyramids: elongated pentagonal pyramid , gyroelongated pentagonal pyramid , pentagonal bipyramid , elongated pentagonal bipyramid , augmented dodecahedron , parabiaugmented dodecahedron , metabiaugmented dodecahedron , and triaugmented dodecahedron . [13] Relatedly, the removal of a pentagonal pyramid from polyhedra is an example known as diminishment; metabidiminished icosahedron and tridiminished icosahedron are the examples in which their constructions begin by removing pentagonal pyramids from a regular icosahedron. [14]

Stereochemistry

In stereochemistry, an atom cluster can have a pentagonal pyramidal geometry. This molecule has a main-group element with one active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory. [15] An example of a molecule with this structure include nido-cage carbonate CB5H9. [16]

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Regular icosahedron</span> Convex polyhedron with 20 triangular faces

In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.

<span class="mw-page-title-main">Kepler–Poinsot polyhedron</span> Any of 4 regular star polyhedra

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

<span class="mw-page-title-main">Truncated icosahedron</span> A polyhedron resembling a soccerball

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Truncated dodecahedron</span> Archimedean solid with 32 faces

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

<span class="mw-page-title-main">Gyroelongated square bipyramid</span> 17th Johnson solid

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

<span class="mw-page-title-main">Triakis icosahedron</span> Catalan solid with 60 faces

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.

<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron, composite polyhedron, and Johnson solid.

<span class="mw-page-title-main">Pentagonal bipyramid</span> Two pentagonal pyramids joined at the bases

In geometry, the pentagonal bipyramid is a polyhedron with 10 triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, composite polyhedron, and Johnson solid.

<span class="mw-page-title-main">Gyroelongated pentagonal pyramid</span> 11th Johnson solid (16 faces)

In geometry, the gyroelongated pentagonal pyramid is a polyhedron constructed by attaching a pentagonal antiprism to the base of a pentagonal pyramid. An alternative name is diminished icosahedron because it can be constructed by removing a pentagonal pyramid from a regular icosahedron.

<span class="mw-page-title-main">Triangular cupola</span> Cupola with hexagonal base

In geometry, the triangular cupola is the cupola with hexagon as its base and triangle as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a cuboctahedron. The triangular cupola can be applied to construct many polyhedrons.

<span class="mw-page-title-main">Snub disphenoid</span> Convex polyhedron with 12 triangular faces

In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape also has alternative names called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron.

<span class="mw-page-title-main">Great dodecahedron</span> Kepler-Poinsot polyhedron

In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedrons. It is composed of 12 pentagonal faces, intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex.

<span class="mw-page-title-main">Small stellated dodecahedron</span> A Kepler-Poinsot polyhedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {52,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.

<span class="mw-page-title-main">Great stellated dodecahedron</span> Kepler–Poinsot polyhedron

In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {52,3}. It is one of four nonconvex regular polyhedra.

References

Notes

  1. 1 2 3 Johnson (1966).
  2. Smith (2000), p.  98.
  3. Pisanski & Servatius (2013), p.  21.
  4. Wohlleben (2019), p.  485–486.
  5. Uehara (2020), p.  62.
  6. Calter & Calter (2011), p.  198.
  7. Berman (1971).
  8. Slobodan, Obradović & Ðukanović (2015).
  9. Rajwade (2001), pp. 8488. See Table 12.3, where denotes the -sided prism and denotes the -sided antiprism.
  10. Gailiunas (2001).
  11. Petrucci, Harwood & Herring (2002), p.  414.
  12. Macartney (2017), p.  482.

Works cited