Regular polygon

Regular polygon

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Set of convex regular p-gons

Regular polygons
Edge
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

s and vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

n
Schläfli symbol {n}
Coxeter–Dynkin diagram
Symmetry group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...

Dn, order 2n
Dual polygon
Dual polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.-Properties:Regular polygons are self-dual....

Self-dual
Area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

(with t=edge length)
Internal angle
Internal angle
In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon...

Internal angle sum
Properties convex
Convex polygon
In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...

, cyclic, equilateral
Equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

, isogonal, isotoxal

A regular polygon is a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

that is equiangular
Equiangular polygon
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon.The only equiangular triangle is the equilateral triangle...

(all angles are equal in measure) and equilateral
Equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...

(all sides have the same length). Regular polygons may be convex or star.

General properties

These properties apply to all regular polygons, whether convex or star.

A regular n-sided polygon has rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

of order n.

All vertices of a regular polygon lie on a common circle (the circumscribed circle
Circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....

), i.e., they are concyclic points
Concyclic points
In geometry, a set of points is said to be concyclic if they lie on a common circle.A circle can be drawn around any triangle. A quadrilateral that can be inscribed inside a circle is said to be a cyclic quadrilateral....

.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the mid-point.

A regular n-sided polygon can be constructed with compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....

if and only if the odd prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

factors of n are distinct Fermat primes. See constructible polygon
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....

.

Symmetry

The symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

of an n-sided regular polygon is dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

Dn (of order 2n): D2, D3
Dihedral group of order 6
The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....

, D4,... It consists of the rotations in Cn, together with reflection symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

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 regular simple polygon
Simple polygon
In geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....

s (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

.

An n-sided convex regular polygon is denoted by its Schläfli symbol {n}.
• Henagon or monogon {1}: degenerate in ordinary space
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

(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
Abstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...

).
• Digon
Digon
In geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...

{2}: a "double line segment": degenerate in ordinary space
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

(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 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:
(or equally of ) degrees,
or full turns
Turn (geometry)
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....

,

and each exterior angle (supplementary to the interior angle) has a measure of degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

Diagonals

For the number of diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...

s is , i.e., 0, 2, 5, 9, ... for a triangle, quadrilateral, 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.

The radius from the centre of a regular polygon to one of the vertices is related to the side length, s or apothem
Apothem
The apothem 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. The word "apothem" can also refer to the length of that line segment. Regular polygons...

, a:

Area

The area A of a convex regular n-sided polygon having side s, apothem
Apothem
The apothem 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. The word "apothem" can also refer to the length of that line segment. Regular polygons...

a, perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

p, and circumradius r is given by

For regular polygons with side s=1, resp. circumradius r=1, resp. apothem a=1, this produces the following table:
SidesNameExact area
(s=1)
Approximate area
(s=1)
Exact area
(r=1)
Approximate area
(r=1))
Approximate area
as fraction of circle
(r=1)
Exact area
(a=1)
Approximate area
(a=1))
Approximate area
as fraction of circle
(a=1)
n regular n-gon
3 equilateral triangle 0.433012702 1.299038105 0.4134966714 5.196152424 1.653986686
4 square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

1.000000000 2.000000000 0.6366197722 4.000000000 1.273239544
5 regular pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

1.720477401   2.377641291 0.7568267288   3.632712640 1.156328347
6 regular hexagon 2.598076211 2.598076211 0.8269933428 3.464101616 1.102657791
7 regular heptagon   3.633912444   2.736410189 0.8710264157   3.371022333 1.073029735
8 regular octagon 4.828427125 2.828427125 0.9003163160   3.313708500 1.054786175
9 regular nonagon   6.181824194   2.892544244 0.9207254290   3.275732109 1.042697914
10 regular decagon
Decagon
In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...

7.694208843   2.938926262 0.9354892840   3.249196963 1.034251515
11 regular hendecagon
Hendecagon
In geometry, a hendecagon is an 11-sided polygon....

9.365639907   2.973524496 0.9465022440   3.229891423 1.028106371
12 regular dodecagon
Dodecagon
In geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...

11.19615242 3.000000000 0.9549296586   3.215390309 1.023490523
13 regular triskaidecagon   13.18576833   3.020700617 0.9615188694   3.204212220 1.019932427
14 regular tetradecagon   15.33450194   3.037186175 0.9667663859   3.195408642 1.017130161
In geometry, a pentadecagon is any 15-sided, 15-angled, polygon.- Regular pentadecagon:A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by...

17.64236291   3.050524822 0.9710122088   3.188348426 1.014882824
In mathematics, a hexadecagon is a polygon with 16 sides and 16 vertices.- Regular hexadecagon :A regular hexadecagon is constructible with a compass and straightedge....

20.10935797   3.061467460 0.9744953584   3.182597878 1.013052368
In geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....

22.73549190   3.070554163 0.9773877456   3.177850752 1.011541311
An octadecagon is a polygon with 18 sides and 18 vertices. Another name for an octadecagon is octakaidecagon.- Construction :A regular octadecagon cannot be constructed using compass and straightedge.- Petrie polygons :...

25.52076819   3.078181290 0.9798155361   3.173885653 1.010279181
In geometry, an enneadecagon is a polygon with 19 sides and angles. It is also known as an enneakaidecagon or a nonadecagon.The radius of the circumcircle of the regular enneadecagon with side length t is...

28.46518943   3.084644958 0.9818729854   3.170539238 1.009213984
20 regular icosagon
Icosagon
In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.One interior angle in a regular icosagon is 162° meaning that one exterior angle would be 18°...

31.56875757   3.090169944 0.9836316430   3.167688806 1.008306663
100 regular hectagon   795.5128988   3.139525977 0.9993421565   3.142626605 1.000329117
1000 regular chiliagon   79577.20975   3.141571983 0.9999934200   3.141602989 1.000003290
10000 regular myriagon   7957746.893   3.141592448 0.9999999345   3.141592757 1.000000033
1,000,000 regular megagon     3.141592654 1.000000000   3.141592654 1.000000000

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

Skew regular polygons

 The cubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... contains a skew regular hexagon, seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis. The zig-zagging side edges of a n-antiprismAntiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles... represent a regular skew 2n-gon, as show in this 17-gonal antiprism.

A regular skew polygon
Skew polygon
In geometry, a skew polygon is a polygon whose vertices do not lie in a plane. Skew polygons must have at least 4 vertices.A regular skew polygon is a skew polygon with equal edge lengths and which is vertex-transitive....

in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

. All edges and internal angles are equal.
 The Platonic solidPlatonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...s (the tetrahedronTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids..., cubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and..., octahedronOctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex...., dodecahedron, and icosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively.

More generally skew regular polygons can be defined in n-space. Examples include the Petrie polygon
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...

s, polygonal paths of edges that divide a regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

into two halves, and seen as a regular polygon in orthogonal projection.

In the infinite limit skew regular polygons become skew apeirogon
Apeirogon
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...

s.

Regular star polygons

A non-convex regular polygon is a regular star polygon. The most common example is the pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...

, which has the same vertices as a pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

, but connects alternating vertices.

For an n-sided star polygon, the Schläfli symbol is modified to indicate the density or 'starriness' m of the polygon, as {n/m}. 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 centre m times.

The (non-degenerate) regular stars of up to 12 sides are:
• Pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...

- {5/2}
• Heptagram
Heptagram
A heptagram or septegram is a seven-pointed star drawn with seven straight strokes.- Geometry :In general, a heptagram is any self-intersecting heptagon ....

- {7/2} and {7/3}
• Octagram
Octagram
In geometry, an octagram is an eight-sided star polygon.- Geometry :In general, an octagram is any self-intersecting octagon ....

- {8/3}
• Enneagram
Enneagram (geometry)
In geometry, an enneagram is a nine-pointed geometric figure. It is sometimes called a nonagram.-Regular enneagram:A regular enneagram is constructed using the same points as the regular enneagon but connected in fixed steps...

- {9/2} and {9/4}
• Decagram - {10/3}
• Hendecagram - {11/2}, {11/3}, {11/4} and {11/5}
• Dodecagram - {12/5}

m and n must be co-prime, or the figure will degenerate.

The degenerate regular stars of up to 12 sides are:
• Hexagram - {6/2}
• Octagram
Octagram
In geometry, an octagram is an eight-sided star polygon.- Geometry :In general, an octagram is any self-intersecting octagon ....

- {8/2}
• Enneagram
Enneagram (geometry)
In geometry, an enneagram is a nine-pointed geometric figure. It is sometimes called a nonagram.-Regular enneagram:A regular enneagram is constructed using the same points as the regular enneagon but connected in fixed steps...

- {9/3}
• Decagram - {10/2} and {10/4}
• Dodecagram - {12/2}, {12/3} and {12/4}

Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example {6/2} 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 {6} to its near neighbours two steps away, to obtain the regular compound of two triangles, or hexagram.
• Many modern geometers, such as Grünbaum (2003), regard this as incorrect. They take the /2 to indicate moving two places around the {6} 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
Abstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...

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

A uniform polyhedron
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

has regular polygons as faces, such that for every two vertices there is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

mapping one into the other (just as there is for a regular polygon).

A quasiregular polyhedron
Quasiregular polyhedron
In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence step closer to regularity than the semiregular which are merely vertex-transitive.There are only two convex...

is a uniform polyhedron which has just two kinds of face alternating around each vertex.

A regular polyhedron
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...

is a uniform polyhedron which has just one kind of face.

The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids.

A polyhedron having regular triangles as faces is called a deltahedron
Deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...

.

• Tiling by regular polygons
Tiling by regular polygons
Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi.- Regular tilings :...

• Platonic solids
• Apeirogon
Apeirogon
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...

- An infinite-sided polygon can also be regular, {∞}.
• List of regular polytopes
• Equilateral polygon