HOME

TheInfoList



OR:

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 axioms ...
, 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 ...
that is direct equiangular (all angles are equal in measure) and equilateral (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 polytop ...
,
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. 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 cons ...
, if the perimeter 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 op ...
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 t ...
(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 (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 ...
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 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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
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 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_iis 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 us ...
.


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 amb ...
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, ...
D''n'' (of order 2''n''): D2, D3, D4, ... It consists of the rotations in C''n'', together with reflection symmetry 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 mo ...
. 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 ; 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 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 ) i ...
(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
diagonal 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 \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 with
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 pol ...
''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(\frac\sum_^n d_i^2 + R^2\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(S^_\right)^m + \sum_^\binom\binomR^\left(S^_ - R^2\right)^k\left(S^_\right)^, and : S^_ = \left(S^_\right)^m + \sum_^\frac\binom\binom \left(S^_ -\left(S^_\right)^2\right)^k\left(S^_\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(R^2+L^2\right)^m+ \sum_^\binom\binomR^L^\left(R^2+L^2\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
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. ...
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 for the ''n'' = 3 case.


Circumradius

The
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 pol ...
''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. ...
''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). ...
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(\sum_^n d_i^2\right)^2 = 2n \sum_^n d_i^4 .


Dissections

Coxeter states that every zonogon (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 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 a ...
''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 pol ...
''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. ...
''a'', and perimeter ''p'' is given by :A = \tfracnsa = \tfracpa = \tfracns^2\cot\left(\tfrac\right) = na^2\tan\left(\tfrac\right) = \tfracnR^2\sin\left(\tfrac\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 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 periods in his '' Disquisitiones Arithmeticae''. 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 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 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 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 ...
'' 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 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 ...
s, polygonal paths of edges that divide a regular polytope 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 t ...
s.


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 operatio ...
. The most common example is the pentagram, 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 mo ...
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 – * Heptagram – 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 Enneagram is a compound word derived from the Greek neoclassical stems for "nine" (''ennea'') and something "written" or "drawn" (''gramma''). Enneagram may refer to: * Enneagram (geometry), a nine-sided star polygon with various configurations ...
– 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 fr ...
– , , and *
Dodecagram In geometry, a dodecagram (γραμμή
Henry George Liddell, Rob ...
– ''m'' and ''n'' must be
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 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-pointe ...
. 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-pointe ...
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 In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruence (geometry), congruent. Unifor ...
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'' ...
mapping one into the other (just as there is for a regular polygon). A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. A regular polyhedron 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.


See also

*
Euclidean tilings by convex regular polygons Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619). Notation of Eu ...
*
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 e ...
*
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 t ...
– An infinite-sided polygon can also be regular, . * List of regular polytopes and compounds * Equilateral polygon * Carlyle circle


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