Edges and vertices | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Schläfli symbol | |||||||||||||||||||||

Coxeter–Dynkin diagram | |||||||||||||||||||||

Symmetry group | D_{n}, order 2n | ||||||||||||||||||||

Dual polygon | Self-dual | ||||||||||||||||||||

Area (with side length ) | |||||||||||||||||||||

Internal angle | |||||||||||||||||||||

Internal angle sum | |||||||||||||||||||||

Inscribed circle diameter | |||||||||||||||||||||

Circumscribed circle diameter | |||||||||||||||||||||

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In Euclidean geometry, a **regular polygon** is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either ** convex **, ** star ** or ** skew **. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.

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

A regular *n*-sided polygon has rotational symmetry 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.

A regular *n*-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of *n* are distinct Fermat primes. See constructible polygon.

A regular *n*-sided polygon can be constructed with origami if and only if for some , where each distinct is a Pierpont prime.^{ [1] }

The symmetry group of an *n*-sided regular polygon is dihedral group D_{n} (of order 2*n*): D_{2}, D_{3}, D_{4}, ... 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.

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 {*n*}. For *n*< 3, we have two degenerate cases:

- Monogon {1}
- 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 {2}; 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.

For a regular convex *n*-gon, each interior angle has a measure of:

- degrees;
- radians; or
- full turns,

and each exterior angle (i.e., 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.

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.

For *n*> 2, the number of diagonals is ; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces OEIS: A007678 .

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*.

For a regular simple *n*-gon with circumradius *R* and distances *d _{i}* from an arbitrary point in the plane to the vertices, we have

For higher powers of distances from an arbitrary point in the plane to the vertices of a regular -gon, if

- ,

then^{ [3] }

- ,

and

- ,

where is a positive integer less than .

If is the distance from an arbitrary point in the plane to the centroid of a regular -gon with circumradius , then^{ [3] }

- ,

where = 1, 2, …, .

For a regular *n*-gon, the sum of the perpendicular distances from any interior point to the *n* sides is *n* times the apothem ^{ [4] }^{: p. 72 } (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the *n* = 3 case.^{ [5] }^{ [6] }

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

For constructible polygons, algebraic expressions 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.^{ [4] }^{: p. 73 }

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.^{ [4] }^{: p.73 }

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} − 1/4*ns*^{2}, where *s* is the side length and *R* is the circumradius.^{ [4] }^{: p. 73 }

If are the distances from the vertices of a regular -gon to any point on its circumcircle, then ^{ [3] }

- .

Coxeter states that every zonogon (a 2*m*-gon whose opposite sides are parallel and of equal length) can be dissected into or 1/2*m*(*m* − 1) parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections *m*-cubes.^{ [7] } 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 OEIS: A006245 gives the number of solutions for smaller polygons.

2m | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 24 | 30 | 40 | 50 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Image | ||||||||||||

Rhombs | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 66 | 105 | 190 | 300 |

The area *A* of a convex regular *n*-sided polygon having side *s*, circumradius *R*, apothem *a*, and perimeter *p* is given by^{ [8] }^{ [9] }

For regular polygons with side *s* = 1, circumradius *R* = 1, or apothem *a* = 1, this produces the following table:^{ [10] } (Note that since as , the area when tends to as grows large.)

Number of sides | Area when side s = 1 | Area when circumradius R = 1 | Area when apothem a = 1 | |||||
---|---|---|---|---|---|---|---|---|

Exact | Approximation | Exact | Approximation | Relative to circumcircle area | Exact | Approximation | Relative to incircle area | |

n | ||||||||

3 | 0.433012702 | 1.299038105 | 0.4134966714 | 5.196152424 | 1.653986686 | |||

4 | 1 | 1.000000000 | 2 | 2.000000000 | 0.6366197722 | 4 | 4.000000000 | 1.273239544 |

5 | 1.720477401 | 2.377641291 | 0.7568267288 | 3.632712640 | 1.156328347 | |||

6 | 2.598076211 | 2.598076211 | 0.8269933428 | 3.464101616 | 1.102657791 | |||

7 | 3.633912444 | 2.736410189 | 0.8710264157 | 3.371022333 | 1.073029735 | |||

8 | 4.828427125 | 2.828427125 | 0.9003163160 | 3.313708500 | 1.054786175 | |||

9 | 6.181824194 | 2.892544244 | 0.9207254290 | 3.275732109 | 1.042697914 | |||

10 | 7.694208843 | 2.938926262 | 0.9354892840 | 3.249196963 | 1.034251515 | |||

11 | 9.365639907 | 2.973524496 | 0.9465022440 | 3.229891423 | 1.028106371 | |||

12 | 11.19615242 | 3 | 3.000000000 | 0.9549296586 | 3.215390309 | 1.023490523 | ||

13 | 13.18576833 | 3.020700617 | 0.9615188694 | 3.204212220 | 1.019932427 | |||

14 | 15.33450194 | 3.037186175 | 0.9667663859 | 3.195408642 | 1.017130161 | |||

15 | ^{ [11] } | 17.64236291 | ^{ [12] } | 3.050524822 | 0.9710122088 | ^{ [13] } | 3.188348426 | 1.014882824 |

16 | ^{ [14] } | 20.10935797 | 3.061467460 | 0.9744953584 | ^{ [15] } | 3.182597878 | 1.013052368 | |

17 | 22.73549190 | 3.070554163 | 0.9773877456 | 3.177850752 | 1.011541311 | |||

18 | 25.52076819 | 3.078181290 | 0.9798155361 | 3.173885653 | 1.010279181 | |||

19 | 28.46518943 | 3.084644958 | 0.9818729854 | 3.170539238 | 1.009213984 | |||

20 | ^{ [16] } | 31.56875757 | ^{ [17] } | 3.090169944 | 0.9836316430 | ^{ [18] } | 3.167688806 | 1.008306663 |

100 | 795.5128988 | 3.139525977 | 0.9993421565 | 3.142626605 | 1.000329117 | |||

1000 | 79577.20975 | 3.141571983 | 0.9999934200 | 3.141602989 | 1.000003290 | |||

10,000 | 7957746.893 | 3.141592448 | 0.9999999345 | 3.141592757 | 1.000000033 | |||

1,000,000 | 79577471545 | 3.141592654 | 1.000000000 | 3.141592654 | 1.000000000 |

Of all *n*-gons with a given perimeter, the one with the largest area is regular.^{ [19] }

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,^{ [20] }^{: p. xi } and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.^{ [20] }^{: pp. 49–50 } 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 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 of the form ) 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.

The cube 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-antiprism represent a regular skew 2n-gon, as shown in this 17-gonal antiprism. |

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.

The Platonic solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) have Petrie polygons, seen in red here, with sides 4, 6, 8, 10, and 10 respectively. |

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

In the infinite limit *regular skew polygons* become skew apeirogons.

2 < 2q < p, gcd(p, q) = 1
| ||||
---|---|---|---|---|

Schläfli symbol | {p/q} | |||

Vertices and Edges | p | |||

Density | q | |||

Coxeter diagram | ||||

Symmetry group | Dihedral (D_{p}) | |||

Dual polygon | Self-dual | |||

Internal angle (degrees) | ^{ [21] } |

A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, 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 center *m* times.

The (non-degenerate) regular stars of up to 12 sides are:

- Pentagram – {5/2}
- Heptagram – {7/2} and {7/3}
- Octagram – {8/3}
- Enneagram – {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 coprime, or the figure will degenerate.

The degenerate regular stars of up to 12 sides are:

- Tetragon – {4/2}
- Hexagons – {6/2}, {6/3}
- Octagons – {8/2}, {8/4}
- Enneagon – {9/3}
- Decagons – {10/2}, {10/4}, and {10/5}
- Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6}

Grünbaum {6/2} or 2{3} ^{ [22] } | Coxeter2{3} or {6}[2{3}]{6} |
---|---|

Doubly-wound hexagon | Hexagram as a compound of two triangles |

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 Coxeter (1948)), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbors two steps away, to obtain the regular compound of two triangles, or hexagram. Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so the hexagram is represented as {6}[2{3}]{6}.
^{ [23] }More compactly Coxeter also writes*2*{n/2}, like*2*{3} 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.^{ [24] } - Many modern geometers, such as Grünbaum (2003),
^{ [22] }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 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.

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.

A uniform polyhedron has regular polygons as faces, such that for every two vertices there is an isometry 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.

- 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
- Carlyle circle

- ↑ Hwa, Young Lee (2017).
*Origami-Constructible Numbers*(PDF) (MA thesis). University of Georgia. pp. 55–59. - ↑ Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
- 1 2 3 Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids".
*Communications in Mathematics and Applications*.**11**: 335–355. - 1 2 3 4 Johnson, Roger A.,
*Advanced Euclidean Geometry*, Dover Publ., 2007 (orig. 1929). - ↑ Pickover, Clifford A,
*The Math Book*, Sterling, 2009: p. 150 - ↑ Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem",
*The College Mathematics Journal*37(5), 2006, pp. 390–391. - ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
- ↑ "Math Open Reference" . Retrieved 4 Feb 2014.
- ↑ "Mathwords".
- ↑ Results for
*R*= 1 and*a*= 1 obtained with Maple, using function definition:The expressions for`f:=proc(n)optionsoperator,arrow;[[convert(1/4*n*cot(Pi/n),radical),convert(1/4*n*cot(Pi/n),float)],[convert(1/2*n*sin(2*Pi/n),radical),convert(1/2*n*sin(2*Pi/n),float),convert(1/2*n*sin(2*Pi/n)/Pi,float)],[convert(n*tan(Pi/n),radical),convert(n*tan(Pi/n),float),convert(n*tan(Pi/n)/Pi,float)]]endproc`

*n*= 16 are obtained by twice applying the tangent half-angle formula to tan(π/4) - ↑
- ↑
- ↑
- ↑
- ↑
- ↑
- ↑
- ↑
- ↑ Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in
*Mathematical Plums*(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147. - 1 2 Bold, Benjamin.
*Famous Problems of Geometry and How to Solve Them*, Dover Publications, 1982 (orig. 1969). - ↑ Kappraff, Jay (2002).
*Beyond measure: a guided tour through nature, myth, and number*. World Scientific. p. 258. ISBN 978-981-02-4702-7. - 1 2 Are Your Polyhedra the Same as My Polyhedra? Branko Grünbaum (2003), Fig. 3
- ↑ Regular polytopes, p.95
- ↑ Coxeter, The Densities of the Regular Polytopes II, 1932, p.53

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*. The bounded plane region, the bounding circuit, or the two together, may be called a polygon.

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 regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, a **hexagon** is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or *circumscribed circle*, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In geometry, a **heptadecagon**, **septadecagon** or **17-gon** is a seventeen-sided polygon.

In geometry, an **octagon** is an eight-sided polygon or 8-gon.

In geometry, a **decagon** is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In geometry, a **diagonal** is a line segment joining two vertices of a polygon or polynomy, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word *diagonal* derives from the ancient Greek διαγώνιος *diagonios*, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as *diagonus*.

In geometry, a **heptagon** or **septagon** is a seven-sided polygon or 7-gon.

In geometry, a **dodecagon** or 12-gon is any twelve-sided polygon.

In geometry, a **triacontagon** or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, a **pentadecagon** or **pentakaidecagon** or 15-gon is a fifteen-sided 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**.

In geometry, a **pyramid** is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a *lateral face*. It is a conic solid with polygonal base. A pyramid with an n-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. All pyramids are self-dual.

In mathematics, a **hexadecagon** is a sixteen-sided polygon.

In geometry, a **bicentric polygon** is a tangential polygon which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.

**Liu Hui's π algorithm** was invented by Liu Hui, a mathematician of the state of Cao Wei. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 or as . Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided π ≈ 142/45 ≈ 3.156. All these empirical π values were accurate to two digits. Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of π to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: π ≈ 3.1416.

In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In Euclidean geometry, an **orthodiagonal quadrilateral** is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In geometry, a **planigon** is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.

- Lee, Hwa Young; "Origami-Constructible Numbers".
- Coxeter, H.S.M. (1948). "Regular Polytopes". Methuen and Co.
`{{cite journal}}`

: Cite journal requires`|journal=`

(help) - 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.

- Weisstein, Eric W. "Regular polygon".
*MathWorld*. - Regular Polygon description With interactive animation
- Incircle of a Regular Polygon With interactive animation
- Area of a Regular Polygon Three different formulae, with interactive animation
- Renaissance artists' constructions of regular polygons at Convergence

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