In

_{''n''} (of order 2''n''): D_{2}, D_{3}, D_{4}, ... It consists of the rotations in C_{''n''}, together with

_{i}'' from an arbitrary point in the plane to the vertices, we have
:$\backslash frac\backslash sum\_^n\; d\_i^4\; +\; 3R^4\; =\; \backslash left(\backslash frac\backslash sum\_^n\; d\_i^2\; +\; R^2\backslash right)^2.$
For higher powers of distances $d\_i$ from an arbitrary point in the plane to the vertices of a regular $n$-gon, if
:$S^\_=\backslash frac\; 1n\backslash sum\_^n\; d\_i^$,
then
:$S^\_\; =\; \backslash left(S^\_\backslash right)^m\; +\; \backslash sum\_^\backslash binom\backslash binomR^\backslash left(S^\_\; -\; R^2\backslash right)^k\backslash left(S^\_\backslash right)^$,
and
:$S^\_\; =\; \backslash left(S^\_\backslash right)^m\; +\; \backslash sum\_^\backslash frac\backslash binom\backslash binom\; \backslash left(S^\_\; -\backslash left(S^\_\backslash right)^2\backslash right)^k\backslash left(S^\_\backslash right)^$,
where $m$ is a positive integer less than $n$.
If $L$ is the distance from an arbitrary point in the plane to the centroid of a regular $n$-gon with circumradius $R$, then
:$\backslash sum\_^n\; d\_i^=n\backslash left(\backslash left(R^2+L^2\backslash right)^m+\; \backslash sum\_^\backslash binom\backslash binomR^L^\backslash left(R^2+L^2\backslash right)^\backslash right)$,
where $m$ = 1, 2, …, $n\; -\; 1$.

^{2} where ''R'' is the circumradius.
The sum of the squared distances from the midpoints of the sides of a regular ''n''-gon to any point on the circumcircle is 2''nR''^{2} − ''ns''^{2}, where ''s'' is the side length and ''R'' is the circumradius.
If $d\_i$ are the distances from the vertices of a regular $n$-gon to any point on its circumcircle, then
:$3\backslash left(\backslash sum\_^n\; d\_i^2\backslash right)^2\; =\; 2n\; \backslash sum\_^n\; d\_i^4$.

Of all ''n''-gons with a given perimeter, the one with the largest area is regular.

Regular Polygon description

With interactive animation

With interactive animation

Three different formulae, with interactive animation

Renaissance artists' constructions of regular polygons

a

Convergence

{{DEFAULTSORT:Regular Polygon Types of polygons

Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, a regular polygon is a polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

that is direct equiangular (all angles are equal in measure) and equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

(all sides have the same length). Regular polygons may be either 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 polytope ...

, star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

or skew
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not ...

. In the limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...

, a sequence of regular polygons with an increasing number of sides approximates a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

, if the perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...

or area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...

is fixed, or a regular apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to th ...

(effectively a straight line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...

), if the edge length is fixed.
General properties

''These properties apply to all regular polygons, whether convex orstar
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

.''
A regular ''n''-sided polygon has rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

of order ''n''.
All vertices of a regular polygon lie on a common circle (the circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every poly ...

.
Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...

.
A regular ''n''-sided polygon can be constructed with compass and straightedge if and only if the odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

factors of ''n'' are distinct Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 429496 ...

s. See constructible polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

.
A regular ''n''-sided polygon can be constructed with origami
) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...

if and only if $n\; =\; 2^\; 3^\; p\_1\; \backslash cdots\; p\_r$ for some $r\; \backslash in\; \backslash mathbb$, where each distinct $p\_i$is a Pierpont prime
In number theory, a Pierpont prime is a prime number of the form
2^u\cdot 3^v + 1\,
for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who use ...

.
Symmetry

Thesymmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

of an ''n''-sided regular polygon is dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...

Dreflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...

in ''n'' axes that pass through the center. If ''n'' is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If ''n'' is odd then all axes pass through a vertex and the midpoint of the opposite side.
Regular convex polygons

All regularsimple polygon
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...

s (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.
An ''n''-sided convex regular polygon is denoted by its Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

. For ''n'' < 3, we have two degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

cases:
; Monogon
In geometry, a monogon, also known as a henagon, is a polygon with one edge and one vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388
In Euclidean geometry
In Eucli ...

: Degenerate in ordinary space. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any abstract polygon.)
; Digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

; a "double line segment": Degenerate in ordinary space. (Some authorities do not regard the digon as a true polygon because of this.)
In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...

must be regular and the faces will be described simply as triangle, square, pentagon, etc.
Angles

For a regular convex ''n''-gon, each interior angle has a measure of: : $\backslash frac$ degrees; : $\backslash frac$ radians; or : $\backslash frac$ full turns, and eachexterior angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

(i.e., supplementary to the interior angle) has a measure of $\backslash tfrac$ degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.
As ''n'' approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon
In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought. Meditation VI by Descartes (English translation).
Regular myriagon
A regular myriag ...

) the internal angle is 179.964°. As the number of sides increase, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with an infinite number of sides.
Diagonals

For ''n'' > 2, the number ofdiagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

s is $\backslash tfracn(n\; -\; 3)$; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces .
For a regular ''n''-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals ''n''.
Points in the plane

For a regular simple ''n''-gon withcircumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

''R'' and distances ''dInterior points

For a regular ''n''-gon, the sum of the perpendicular distances from any interior point to the ''n'' sides is ''n'' times theapothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...

for the ''n'' = 3 case.
Circumradius

Thecircumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

''R'' from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

''a'' by
:$R\; =\; \backslash frac\; =\; \backslash frac\; \backslash quad\_,\backslash quad\; a\; =\; \backslash frac$
For constructible polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

s, algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). ...

s for these relationships exist; see Bicentric polygon#Regular polygons.
The sum of the perpendiculars from a regular ''n''-gon's vertices to any line tangent to the circumcircle equals ''n'' times the circumradius.
The sum of the squared distances from the vertices of a regular ''n''-gon to any point on its circumcircle equals 2''nR''Dissections

Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...

states that every zonogon
In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
Examples
A regular polygon is a zonogon if and ...

(a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into $\backslash tbinom$ or parallelograms.
These tilings are contained as subsets of vertices, edges and faces in orthogonal projections ''m''-cubes.
In particular, this is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi.
The list gives the number of solutions for smaller polygons.
Area

The area ''A'' of a convex regular ''n''-sided polygon havingside
Side or Sides may refer to:
Geometry
* Edge (geometry) of a polygon (two-dimensional shape)
* Face (geometry) of a polyhedron (three-dimensional shape)
Places
* Side (Ainis), a town of Ainis, ancient Thessaly, Greece
* Side (Caria), a town of an ...

''s'', circumradius
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

''R'', apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...

''a'', and perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...

''p'' is given by
:$A\; =\; \backslash tfracnsa\; =\; \backslash tfracpa\; =\; \backslash tfracns^2\backslash cot\backslash left(\backslash tfrac\backslash right)\; =\; na^2\backslash tan\backslash left(\backslash tfrac\backslash right)\; =\; \backslash tfracnR^2\backslash sin\backslash left(\backslash tfrac\backslash right)$
For regular polygons with side ''s'' = 1, circumradius ''R'' = 1, or apothem ''a'' = 1, this produces the following table: (Note that since $\backslash cot\; x\; \backslash rightarrow\; 1/x$ as $x\; \backslash rightarrow\; 0$, the area when $s\; =\; 1$ tends to $n^2/4\backslash pi$ as $n$ grows large.)
Constructible polygon

Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. Theancient Greek mathematicians
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969). This led to the question being posed: is it possible to construct ''all'' regular ''n''-gons with compass and straightedge? If not, which ''n''-gons are constructible and which are not?
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier tra ...

s in his ''Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...

''. This theory allowed him to formulate a sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for the constructibility of regular polygons:
: A regular ''n''-gon can be constructed with compass and straightedge if ''n'' is the product of a power of 2 and any number of distinct Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 429496 ...

s (including none).
(A Fermat prime is a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

of the form $2^\; +\; 1.$) Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
In a paper from 1837, Wantzel pr ...

in 1837. The result is known as the Gauss–Wantzel theorem.
Equivalently, a regular ''n''-gon is constructible if and only if the cosine of its common angle is a constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...

—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.
Regular skew polygons

A ''regularskew polygon
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not ...

'' in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...

. All edges and internal angles are equal.
More generally ''regular skew polygons'' can be defined in ''n''-space. Examples include the Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

s, polygonal paths of edges that divide a regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

into two halves, and seen as a regular polygon in orthogonal projection.
In the infinite limit ''regular skew polygons'' become skew apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to th ...

s.
Regular star polygons

A non-convex regular polygon is a regular star polygon. The most common example is thepentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

, which has the same vertices as a pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...

, but connects alternating vertices.
For an ''n''-sided star polygon, the Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

is modified to indicate the ''density'' or "starriness" ''m'' of the polygon, as . If ''m'' is 2, for example, then every second point is joined. If ''m'' is 3, then every third point is joined. The boundary of the polygon winds around the center ''m'' times.
The (non-degenerate) regular stars of up to 12 sides are:
*Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

–
*Heptagram
A heptagram, septagram, septegram or septogram is a seven-point star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek suffix ''-gram''. The ''-gram'' suffix derives from ''γρ ...

– and
*Octagram
In geometry, an octagram is an eight-angled star polygon.
The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line".
Deta ...

–
* Enneagram – and
* Decagram –
*Hendecagram
In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices.
The name ''hendecagram'' combines a Greek numeral prefix, '' hendeca-'', with the Greek suffix ''-gram''. The ''hendeca-'' prefix derives fro ...

– , , and
*Dodecagram
In geometry, a dodecagram (γραμμή

Henry George Liddell, Robe ...

–
''m'' and ''n'' must be Henry George Liddell, Robe ...

coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

, or the figure will degenerate.
The degenerate regular stars of up to 12 sides are:
*Tetragon –
*Hexagons – ,
*Octagons – ,
*Enneagon –
*Decagons – , , and
*Dodecagons – , , , and
Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example, may be treated in either of two ways:
* For much of the 20th century (see for example ), we have commonly taken the /2 to indicate joining each vertex of a convex to its near neighbors two steps away, to obtain the regular compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struc ...

of two triangles, or hexagram
, can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green).
A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...

. Coxeter clarifies this regular compound with a notation for the compound , so the hexagram
, can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green).
A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...

is represented as More compactly Coxeter also writes ''2'', like ''2'' for a hexagram as compound as alternations of regular even-sided polygons, with italics on the leading factor to differentiate it from the coinciding interpretation.Coxeter, The Densities of the Regular Polytopes II, 1932, p.53
* Many modern geometers, such as Grünbaum (2003), regard this as incorrect. They take the /2 to indicate moving two places around the at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories of abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...

s, but it also more closely copies the way in which Poinsot (1809) created his star polygons – by taking a single length of wire and bending it at successive points through the same angle until the figure closed.
Duality of regular polygons

All regular polygons are self-dual to congruency, and for odd ''n'' they are self-dual to identity. In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.Regular polygons as faces of polyhedra

Auniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...

has regular polygons as faces, such that for every two vertices there is an isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

mapping one into the other (just as there is for a regular polygon).
A quasiregular polyhedron
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the se ...

is a uniform polyhedron which has just two kinds of face alternating around each vertex.
A regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

is a uniform polyhedron which has just one kind of face.
The remaining (non-uniform) convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

with regular faces are known as the Johnson solids
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson ...

.
A polyhedron having regular triangles as faces is called a deltahedron
In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many d ...

.
See also

*Euclidean tilings by convex regular polygons
Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...

*Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

*Apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to th ...

– An infinite-sided polygon can also be regular, .
*List of regular polytopes and compounds
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...

*Equilateral polygon
In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon ...

*Carlyle circle
In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of ...

Notes

References

*Lee, Hwa Young; "Origami-Constructible Numbers". * *Grünbaum, B.; Are your polyhedra the same as my polyhedra?, ''Discrete and comput. geom: the Goodman-Pollack festschrift'', Ed. Aronov et al., Springer (2003), pp. 461–488. * Poinsot, L.; Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' 9 (1810), pp. 16–48.External links

*Regular Polygon description

With interactive animation

With interactive animation

Three different formulae, with interactive animation

Renaissance artists' constructions of regular polygons

a

Convergence

{{DEFAULTSORT:Regular Polygon Types of polygons