Regular polygon

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Regular polygon

A regular polygon is a polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
which is equiangular

Equiangular polygon

File:Rectangle definition.svgIn 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....
(all angles are congruent

Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
) 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

Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
.

General properties
These properties apply to both convex and a star regular polygons.

All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points

Concyclic points

In geometry, a set of point is said to be concyclic if they lie on a common circle.A circumscribed circle any triangle. A quadrilateral that can be inscribed inside a circle is said to be a cyclic quadrilateral....
, i.e., every regular polygon has a 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....
.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle.

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 or angles using only an Idealization ruler and Compass ....
if and only if the odd prime

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
factors of n are distinct Fermat primes.

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General properties

These properties apply to both convex and a star regular polygons.

All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points

Concyclic points

Circumscribed circle

Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization ruler and Compass ....
if and only if the odd prime

Prime number

Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions. 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 Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
of an n-sided regular polygon is dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
D_n (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 D'3 and the symmetric group of degree 3, with notation S'3....
, D₄

Examples of groups

Some elementary examples of groups in mathematics are given on Group .Further examples are listed here....
,... It consists of the rotations in C_n (there is rotational symmetry

Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation....
of order n), together with reflection symmetry

Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
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 closed polygonal chain of line segments that do not cross each other. That is, it consists of finitely many line segments, each line segment endpoint is shared by two segments, and the segments do not otherwise intersect....
s (a simple polygon is one which 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. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
.

An n-sided convex regular polygon is denoted by its Schläfli symbol .

Henagon
Henagon

In geometry a henagon is a polygon with one Edge and one Vertex . It has Schl?fli symbol . Since a henagon has only one side and only one interior angle, every henagon is regular polygon by definition....
or monogon : degenerate in ordinary space
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
(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 combinatorics 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 Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....
: a "double line segment": degenerate in ordinary space
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
(Some authorities do not regard the digon as a true polygon because of this).
Equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
Regular tetragon or quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
including square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
Regular pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
Regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
Regular heptagon
Heptagon

In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....
Regular octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
Regular enneagon
Enneagon

In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a hybrid word, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century....
or nonagon
Regular decagon
Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....
Regular hendecagon
Hendecagon

In geometry, a hendecagon is an 11-sided polygon.The name "undecagon" is often seen as incorrect, but the matter is up for debate. The Greek language prefix 'hen', is preferable to the Latin 'uni' or 'un' ....
Regular dodecagon
Dodecagon

In geometry, a dodecagon is any polygon with 12 sides and twelve angles....
Regular triskaidecagon
Triskaidecagon

In geometry, a triskaidecagon is a polygon with 13 sides and angles.The measure of each internal angle of a Regular polygon triskaidecagon is approximately 152.308 degree s, and the area with side length a is given by...
Regular tetrakaidecagon

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 radians,

or full turns

Turn (geometry)

A turn is a unit of plane angle, equal to 360? or 2p radians. As an angular unit it is mainly useful for large angles, such as in connection with coils and rotation objects....
,

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 can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
s is , i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.

Area

The area A of a convex regular n-sided polygon having sides of length t is:

in degrees

or in radians

If the circumradius r (length of the segment joining the center to the vertex) is known, the area is:

in degrees

or in radians ,

Also, the area is half the perimeter

Semiperimeter

In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name....
multiplied by the length of the 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....
, a, (the line drawn from the centre of the polygon perpendicular to a side). That is A = a.n.t/2, as the length of the perimeter is n.t, or more simply 1/2 p.a.

For sides t=1 this gives:

in degrees

or in radians (n not equal to 2)

with the following values:

Sides	Name	Exact area	Approximate area
3	equilateral triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....		0.433
4	square Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....	1	1.000
5	regular-pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....		1.720
6	regular-hexagon Hexagon In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....		2.598
7	regular-heptagon Heptagon In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....		3.634
8	regular-octagon Octagon In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....		4.828
9	regular-enneagon Enneagon In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a hybrid word, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century....		6.182
10	regular-decagon Decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....		7.694
11	regular-hendecagon Hendecagon In geometry, a hendecagon is an 11-sided polygon.The name "undecagon" is often seen as incorrect, but the matter is up for debate. The Greek language prefix 'hen', is preferable to the Latin 'uni' or 'un' ....		9.366
12	regular-dodecagon Dodecagon In geometry, a dodecagon is any polygon with 12 sides and twelve angles....		11.196
13	regular-triskaidecagon Triskaidecagon In geometry, a triskaidecagon is a polygon with 13 sides and angles.The measure of each internal angle of a Regular polygon triskaidecagon is approximately 152.308 degree s, and the area with side length a is given by...		13.186
14	regular-tetradecagon Tetradecagon In geometry, a tetrakaidecagon is a polygon with 14 sides and angles.The area of a Regular polygon tetradecagon of side length a is given by...		15.335
15	regular-pentadecagon Pentadecagon In geometry, a pentadecagon is any 15-sided, 15-angled, polygon.A Regular polygon pentadecagon has interior angles of 156?, and with a side length a, has an area given by...		17.642
16	regular-hexadecagon Hexadecagon In mathematics, a hexadecagon is a polygon with 16 Edge and 16 Vertex .A regular hexadecagon is constructible polygon with a Compass and straightedge constructions....		20.109
17	regular-heptadecagon Heptadecagon In geometry, a heptadecagon is a seventeen-sided polygon....		22.735
18	regular-octadecagon Octadecagon An octadecagon is a polygon with 18 Edge and 18 Vertex . Another name for an octadecagon is octakaidecagon....		25.521
19	regular-enneadecagon Enneadecagon 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...		28.465
20	regular-icosagon Icosagon In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.As a golygonal path, the swastika is considered to be an irregular icosagon....		31.569
100	regular-hectagon		795.513
1000	regular-chiliagon		79577.210
10000	regular-myriagon		7957746.893

The amounts that the areas are less than those of circles with the same perimeter

Perimeter

A perimeter is a path that bounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length....
, are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing n to the limit p/12).

Regular star polygons

A non-convex regular polygon is a regular star polygon

Star polygon

Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
, 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 internal angles in a simple pentagon total 540?....
, but connects alternating vertices.

For an n-sided star polygon, the Schläfli symbol is modified to indicate the '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 centre m times, and m is sometimes called the density of the polygon.

Examples:

Pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
-
Hexagram
Hexagram

A hexagram is a six-pointed geometric star figure, or 2, the compound of two equilateral triangle s. The intersection is a regular hexagon.While generally recognized as a symbol of Jewish identity it is used also in other historical, religious and cultural contexts, for example in #Use of the Star by Arabs and Muslims, and #Occurrence in...
-
Heptagram
Heptagram

A heptagram or septegram is a seven-pointed Star drawn with seven straight strokes....
- and
Octagram
Octagram

In geometry, an octagram is an eight-sided star polygon....
-
Enneagram
Enneagram

In geometry, an enneagram is a nine-pointed geometric figure. The term derives from two ancient Greek words: ennea and gramma ....
- and
Decagram
Decagram (geometry)

In geometry, a decagram is a 10-sided star polygon.There is one regular decagram star polygon, , containing the vertices of a regular decagon, but connected by every third point....
-
Hendecagram
Hendecagram

A hendecagram is a star polygon that has eleven Point . There are 4 regular forms: , , , ....
- , , ,
Triskaidecagram-, , , ,

m and n must be co-prime, or the figure will degenerate. 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 neighbours two steps away, to obtain the regular compound of two triangles, or hexagram
Hexagram

A hexagram is a six-pointed geometric star figure, or 2, the compound of two equilateral triangle s. The intersection is a regular hexagon.While generally recognized as a symbol of Jewish identity it is used also in other historical, religious and cultural contexts, for example in #Use of the Star by Arabs and Muslims, and #Occurrence in...
.
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
Abstract polytope

In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics 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 polygon

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s are self-dual to congruency, and for even 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 polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
has regular polygons as faces, such that for every two vertices there is an isometry

Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
mapping one into the other (just as there is for a regular polygon).

A quasiregular polyhedron

Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
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 Congruence 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....
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 polygon having regular triangles as faces is called a deltahedron

Deltahedron

A deltahedron is a polyhedron whose face s are all equilateral triangles. The name is taken from the Greek language majuscule delta , which has the shape of an equilateral triangle....
.

External links

With interactive animation
With interactive animation
Three different formulae, with interactive animation
at

Regular polygon

General properties

Symmetry

Regular convex polygons

Angles

Diagonals

Area

Regular star polygons

Duality of regular polygons

Regular polygons as faces of polyhedra

See also

External links