In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a pentagonal pyramid is a
pyramid
A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
with a
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
base and five triangular faces, having a total of six faces. It is categorized as a
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
if all of the edges are equal in length, forming
equilateral triangular faces and a
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
al base.
Pentagonal pyramids occur as pieces and tools in the construction of many polyhedra. They also appear in the field of
natural science
Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer ...
, as in
stereochemistry
Stereochemistry, a subdiscipline of chemistry, studies the spatial arrangement of atoms that form the structure of molecules and their manipulation. The study of stereochemistry focuses on the relationships between stereoisomers, which are defined ...
where the shape can be described as the
pentagonal pyramidal molecular geometry
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, as well as the study of
shell
Shell may refer to:
Architecture and design
* Shell (structure), a thin structure
** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses
Science Biology
* Seashell, a hard outer layer of a marine ani ...
assembling in the underlying
potential energy surface
A potential energy surface (PES) or energy landscape describes the energy of a Physical system, system, especially a collection of atoms, in terms of certain Parameter, parameters, normally the positions of the atoms. The Surface (mathematics), ...
s and
disclination
In crystallography, a disclination is a line defect in which there is compensation of an angular gap. They were first discussed by Vito Volterra in 1907, who provided an analysis of the elastic strains of a wedge disclination. By analogy to disloc ...
in
fiveling
A fiveling, also known as a decahedral nanoparticle, a multiply-twinned particle (MTP), a pentagonal nanoparticle, a pentatwin, or a five-fold twin is a type of twinned crystal that can exist at sizes ranging from nanometers to millimetres. It ...
s and related shapes such as pyramidal
copper
Copper is a chemical element; it has symbol Cu (from Latin ) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish-orang ...
and other metal
nanowires
upright=1.2, Crystalline 2×2-atom tin selenide nanowire grown inside a single-wall carbon nanotube (tube diameter ≈1 nm).
A nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 m). Mor ...
.
Properties
A pentagonal pyramid has six vertices, ten edges, and six faces. One of its faces is
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, a ''base'' of the pyramid; five others are
triangles
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...
. 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
The apex is the highest point of something. The word may also refer to:
Arts and media Fictional entities
* Apex (comics)
A-Bomb
Abomination
Absorbing Man
Abraxas
Abyss
Abyss is the name of two characters appearing in Ameri ...
. A pentagonal pyramid is said to be ''regular'' if its base is
circumscribed in a circle that forms a
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, and it is said to be ''right'' if its altitude is erected perpendicularly to the base's center.
Like other right pyramids with a regular polygon as a base, this pyramid has
pyramidal symmetry
In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object.
They are ...
of
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
: the pyramid is left invariant by rotations of one, two, three, four-fifths around its
axis of symmetry
An axis (: axes) may refer to:
Mathematics
*A specific line (often a directed line) that plays an important role in some contexts. In particular:
** Coordinate axis of a coordinate system
*** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
, 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. It can be represented as the
wheel graph
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with vertices can also be defined as the 1-skeleton of an pyramid.
Some authors write to denote a wheel graph ...
, meaning its
skeleton
A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...
can be interpreted as a pentagon in which its five vertices connects a vertex in the center called the
universal vertex
In graph theory, a universal vertex is a Vertex (graph theory), vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. A ...
. It is
self-dual, meaning its
dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...
is the pentagonal pyramid itself.

When all edges are equal in length, the five triangular faces are
equilateral
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
and the base is a regular pentagon. Because this pyramid remains
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and all of its faces are
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s, it is classified as the second
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
. The
dihedral angle between two adjacent triangular faces is approximately 138.19° and that between the triangular face and the base is 37.37°. It is an
elementary polyhedron
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids.
Definition and examples
A convex polyhedron is said to be ''composite'' if there exists ...
, meaning that it cannot be separated by a plane to create two small convex polyhedrons with regular faces. A
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
's
surface area
The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
is the sum of the areas of its faces. Therefore, the surface area of a pentagonal pyramid is the sum of the areas of the five triangles and the one pentagon. The volume of every pyramid equals one-third of the area of its base multiplied by its height. So, the volume of a pentagonal pyramid is one-third of the product of the height and a pentagonal pyramid's area. In the case of Johnson solid with edge length
, its surface area
and volume
are:
Applications

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
antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s is known as
elongation or
gyroelongation, respectively. Examples of polyhedrons are the
pentakis dodecahedron
In geometry, a pentakis dodecahedron or kisdodecahedron is a polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. Specifically, the term typically refers to ...
is constructed from the
dodecahedron
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 Kepler–Po ...
by attaching the base of pentagonal pyramids onto each pentagonal face,
small stellated dodecahedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
is constructed from a
regular dodecahedron
A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...
stellated by pentagonal pyramids, and a
regular icosahedron
The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...
constructed from a
pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of ten triangles fo ...
by attaching two pentagonal pyramids onto its pentagonal bases. Some Johnson solids are constructed by either augmenting pentagonal pyramids or augmenting other shapes with pentagonal pyramids: an
elongated pentagonal pyramid , a
gyroelongated pentagonal pyramid
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 pe ...
, a
pentagonal bipyramid
The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten 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 ...
, an
elongated pentagonal bipyramid , an
augmented dodecahedron , a
parabiaugmented dodecahedron , a
metabiaugmented dodecahedron , and a
triaugmented dodecahedron .
[, pp]
84–88
See Table 12.3, where denotes the prism and denotes the antiprism. Relatedly, the removal of a pentagonal pyramid from polyhedra is an example of a technique known as
diminishment Diminishment is the legal process by which the United States Congress can reduce the size of an Indian reservation.
History
In 1984, the United States Supreme Court
The Supreme Court of the United States (SCOTUS) is the highest court in the ...
; the
metabidiminished icosahedron
In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron, replacing two sets of five triangular faces of the ic ...
and
tridiminished icosahedron
In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron.
Construction
The tridiminished icosahedron can be constructed by removing three regular pentago ...
are the examples in which their constructions begin by removing pentagonal pyramids from a regular icosahedron.
In
stereochemistry
Stereochemistry, a subdiscipline of chemistry, studies the spatial arrangement of atoms that form the structure of molecules and their manipulation. The study of stereochemistry focuses on the relationships between stereoisomers, which are defined ...
, an
atom cluster
Nanoclusters are atomically precise, crystalline materials most often existing on the 0-2 nanometer scale. They are often considered kinetically stable intermediates that form during the synthesis of comparatively larger materials such as semic ...
can have a
pentagonal pyramidal geometry. This molecule has a main-group element with one active
lone pair
In chemistry, a lone pair refers to a pair of valence electrons that are not shared with another atom in a covalent bondIUPAC ''Gold Book'' definition''lone (electron) pair''/ref> and is sometimes called an unshared pair or non-bonding pair. Lone ...
of
electrons
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, which can be described by a model that predicts the geometry of molecules known as
VSEPR theory
Valence shell electron pair repulsion (VSEPR) theory ( , ) is a conceptual model, model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. It is also named the Gill ...
. An example of a molecule with this structure is
nido-cage
carbonate
A carbonate is a salt of carbonic acid, (), characterized by the presence of the carbonate ion, a polyatomic ion with the formula . The word "carbonate" may also refer to a carbonate ester, an organic compound containing the carbonate group ...
CB
5H
9.
The formation of
virus
A virus is a submicroscopic infectious agent that replicates only inside the living Cell (biology), cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Viruses are ...
shells, known as
capsids
A capsid is the protein shell of a virus, enclosing its genetic material. It consists of several oligomeric (repeating) structural subunits made of protein called Protomer (structural biology), protomers. The observable 3-dimensional morpholog ...
, can be modeled from pieces shaped like pentagonal and hexagonal pyramids. These shapes were chosen to resemble those of the protein subunits of natural viruses. By appropriately choosing the attractive and repulsive forces between pyramids, they found that the pyramids could self-assemble into icosahedral shells reminiscent of those found in nature.
The
relaxation of internal elastic
stress fields due to
disclination
In crystallography, a disclination is a line defect in which there is compensation of an angular gap. They were first discussed by Vito Volterra in 1907, who provided an analysis of the elastic strains of a wedge disclination. By analogy to disloc ...
s in twinned
copper
Copper is a chemical element; it has symbol Cu (from Latin ) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkish-orang ...
particles. Such a shape is the pentagonal pyramid, which allows growth to a large size and preserves symmetry. This can be done by activating
cathode
A cathode is the electrode from which a conventional current leaves a polarized electrical device such as a lead-acid battery. This definition can be recalled by using the mnemonic ''CCD'' for ''Cathode Current Departs''. Conventional curren ...
by the process of initial crystal growth in the
electrolyte
An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
, by the movement of
aluminum
Aluminium (or aluminum in North American English) is a chemical element; it has chemical symbol, symbol Al and atomic number 13. It has a density lower than that of other common metals, about one-third that of steel. Aluminium has ...
and
silicon oxide Silicon oxide may refer to either of the following:
*Silicon dioxide
Silicon dioxide, also known as silica, is an oxide of silicon with the chemical formula , commonly found in nature as quartz. In many parts of the world, silica is the maj ...
s' abrasive particles.
References
Notes
Works cited
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External links
*
Virtual Reality Polyhedrawww.georgehart.com: The Encyclopedia of Polyhedra (
VRML
VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...
br>
model
{{Johnson solids navigator
Elementary polyhedron
Johnson solids
Prismatoid polyhedra
Pyramids (geometry)
Self-dual polyhedra