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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, 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 t ...
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 ...
(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 polyto ...
,
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Eart ...
or skew. In the limit, 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 con ...
, 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 prac ...
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 or planar lamina, while '' surface area'' refers to the area of an open ...
is fixed, or a regular apeirogon (effectively a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
), if the edge length is fixed.

# General properties

''These properties apply to all regular polygons, whether convex or
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Eart ...
.'' 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 ...
of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. 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 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 p ...
. 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 ...
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 way ...
factors of ''n'' are distinct Fermat primes. See constructible polygon. A regular ''n''-sided polygon can be constructed with origami if and only if $n = 2^ 3^ p_1 \cdots p_r$ for some $r \in \mathbb$, where each distinct $p_i$is a Pierpont prime.

## Symmetry

The
symmetry 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 ambi ...
of an ''n''-sided regular polygon is dihedral group D''n'' (of order 2''n''): D2, D3, D4, ... It consists of the rotations in C''n'', together with
reflection 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 th ...
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 polygons (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 mor ...
. For ''n'' < 3, we have two degenerate cases: ; Monogon : 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 vis ...
; 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 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: : $\frac$ degrees; : $\frac$ radians; or : $\frac$ full turns, and each exterior angle (i.e., supplementary to the interior angle) has a measure of $\tfrac$ 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) 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 of diagonals is $\tfracn\left(n - 3\right)$; 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 with circumradius ''R'' and distances ''di'' from an arbitrary point in the plane to the vertices, we have :$\frac\sum_^n d_i^4 + 3R^4 = \left\left(\frac\sum_^n d_i^2 + R^2\right\right)^2.$ For higher powers of distances $d_i$ from an arbitrary point in the plane to the vertices of a regular $n$-gon, if :$S^_=\frac 1n\sum_^n d_i^$, then :$S^_ = \left\left(S^_\right\right)^m + \sum_^\binom\binomR^\left\left(S^_ - R^2\right\right)^k\left\left(S^_\right\right)^$, and :$S^_ = \left\left(S^_\right\right)^m + \sum_^\frac\binom\binom \left\left(S^_ -\left\left(S^_\right\right)^2\right\right)^k\left\left(S^_\right\right)^$, 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 :$\sum_^n d_i^=n\left\left(\left\left(R^2+L^2\right\right)^m+ \sum_^\binom\binomR^L^\left\left(R^2+L^2\right\right)^\right\right)$, where $m$ = 1, 2, …, $n - 1$.

### Interior points

For a regular ''n''-gon, the sum of the perpendicular distances from any interior point to the ''n'' sides is ''n'' times the apothemJohnson, 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.

The circumradius ''R'' from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the apothem ''a'' by :$R = \frac = \frac \quad_,\quad a = \frac$ For constructible polygons,
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). F ...
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''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 :$3\left\left(\sum_^n d_i^2\right\right)^2 = 2n \sum_^n d_i^4$.

## Dissections

Coxeter 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 an ...
(a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into $\tbinom$ 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 having side ''s'', circumradius ''R'', apothem ''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 prac ...
''p'' is given by :$A = \tfracnsa = \tfracpa = \tfracns^2\cot\left\left(\tfrac\right\right) = na^2\tan\left\left(\tfrac\right\right) = \tfracnR^2\sin\left\left(\tfrac\right\right)$ For regular polygons with side ''s'' = 1, circumradius ''R'' = 1, or apothem ''a'' = 1, this produces the following table: (Note that since $\cot x \rightarrow 1/x$ as $x \rightarrow 0$, the area when $s = 1$ tends to $n^2/4\pi$ as $n$ grows large.)
Of all ''n''-gons with a given perimeter, the one with the largest area is regular.

# Constructible polygon

Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all. The ancient Greek mathematicians 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 refe ...
proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his '' Disquisitiones Arithmeticae''. This theory allowed him to formulate a sufficient condition 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 primes (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 (mathematics), 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 ...
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 pro ...
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—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.

# Regular skew polygons

A ''regular skew polygon'' in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism. All edges and internal angles are equal. More generally ''regular skew polygons'' can be defined in ''n''-space. Examples include the Petrie polygons, 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 apeirogons.

# Regular star polygons

A non-convex regular polygon is a regular
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operation ...
. The most common example is the
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 ...
, 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 sim ...
, 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 mor ...
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 – and * Octagram – * Enneagram – and * Decagram – * Hendecagram – , , and * Dodecagram – ''m'' and ''n'' must be coprime, 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 struct ...
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 polytopes, 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 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 ...
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 equiv ...
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.

* Euclidean tilings by convex regular polygons * Platonic solid * Apeirogon – An infinite-sided polygon can also be regular, . * List of regular polytopes and compounds *
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 polyg ...
* Carlyle circle

# 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.