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Set of regular p-gons | |||
---|---|---|---|

| |||

Edges and vertices | p | ||

Schläfli symbol | {p} | ||

Coxeter–Dynkin diagram | - | Symmetry group | Dihedral (D_{p}) |

Dual polyhedron | Self-dual | ||

Area (with t=edge length) | $A=frac\{pt^2\}\{4tan(pi/p)\}$ | ||

Internal angle (degrees) | $left(1-frac\{2\}\{p\}right)times\; 180$ | ||

Internal angle sum (degrees) | $left(p-2right)times\; 180$ |

These properties apply to both convex and star regular polygons.

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

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 if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

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

- Henagon or monogon: degenerate in ordinary space {1}
- Digon: a "double line segment" - degenerate in ordinary space {2}
- Equilateral triangle {3}
- Square {4}
- Regular pentagon {5}
- Regular hexagon {6}
- Regular heptagon {7}
- Regular octagon {8}
- Regular enneagon or nonagon {9}
- Regular decagon {10}
- Regular hendecagon {11}
- Regular dodecagon {12}
- Regular triskaidecagon {13}
- Regular tetrakaidecagon {14}

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:

- $(1-frac\{2\}\{n\})times\; 180$ (or equally of $(n-2)times\; frac\{180\}\{n\}$ ) degrees,

- or $frac\{(n-2)pi\}\{n\}$ radians,

- or $frac\{(n-2)\}\{2n\}$ full turns,

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

For $n\; >\; 2$ the number of diagonals is $frac\{n\; (n-3)\}\{2\}$, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.

The area A of a convex regular n-sided polygon is:

in degrees

- $A=frac\{nt^2\}\{4tan(frac\{180\}\{n\})\}$,

or in radians

- $A=frac\{nt^2\}\{4tan(frac\{pi\}\{n\})\}$,

where t is the length of a side.

If the radius, or length of the segment joining the center to the vertex is known, the area is:

in degrees

- $A=frac\{nr^2sin(frac\{360\}\{n\})\}\{2\}$

or in radians

- $A=frac\{nr^2sin(frac\{2\; pi\}\{n\})\}\{2\}$,

where r is the radius

Also, the area is half the perimeter multiplied by the length of the apothem, 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

- $frac\{n\}\{4tan(frac\{180\}\{n\})\}$

or in radians (n not equal to 2)

- $\{frac\{n\}\{4\}\}\; cot(pi/n)$

with the following values:

Sides | Name | Exact area | Approximate area |
---|---|---|---|

3 | equilateral triangle | $frac\{sqrt\{3\}\}\{4\}$ | 0.433 |

4 | square | 1 | 1.000 |

5 | regular-pentagon | $frac\; \{1\}\{4\}\; sqrt\{25+10sqrt\{5\}\}$ | 1.720 |

6 | regular-hexagon | $frac\{3\; sqrt\{3\}\}\{2\}$ | 2.598 |

7 | regular-heptagon | 3.634 | |

8 | regular-octagon | $2\; +\; 2\; sqrt\{2\}$ | 4.828 |

9 | regular-enneagon | 6.182 | |

10 | regular-decagon | $frac\{5\}\{2\}\; sqrt\{5+2sqrt\{5\}\}$ | 7.694 |

11 | regular-hendecagon | 9.366 | |

12 | regular-dodecagon | $6+3sqrt\{3\}$ | 11.196 |

13 | regular-triskaidecagon | 13.186 | |

14 | regular-tetradecagon | 15.335 | |

15 | regular-pentadecagon | 17.642 | |

16 | regular-hexadecagon | 20.109 | |

17 | regular-heptadecagon | 22.735 | |

18 | regular-octadecagon | 25.521 | |

19 | regular-enneadecagon | 28.465 | |

20 | regular-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, are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing n to the limit π/12).

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 'starriness' m of the polygon, as {n/m}. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of the polygon winds around the centre m times, and m is sometimes called the density of the polygon.

Examples:

- 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}, {11/5}

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 {6/2} may be treated in either of two ways:

- For much of the 20th century (see for example ), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbours two steps away, to obtain the regular compound of two triangles, or hexagram.
- Many modern geometers, such as Grünbaum (2003), regard this as incorrect. They take the /2 to indicate moving two places around the {6} at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories of abstract 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.

In addition the regular star polygons and regular star figures (compounds), being composed of regular polygons, are also self-dual.

The remaining convex polyhedra with regular faces are known as the Johnson solids.

- 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.

- Tiling by regular polygons
- Platonic solids
- Apeirogon - An infinite-sided polygon can also be regular, {∞}.
- List of regular polytopes
- Equilateral polygon

- 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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Last updated on Friday October 10, 2008 at 12:55:50 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 10, 2008 at 12:55:50 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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