View source for Regular polygon

{{Regular polygon db|Regular polygon stat table|p17}}
A '''regular polygon''' is a [[shape]] that can be drawn on a flat surface. It has [[Side|sides]] that are all the same [[length]], and its [[Angle|angles]] are all the same. In other words, a [[polygon]] which is both [[equilateral polygon|equilateral]] (which means all its sides have the same length) and [[equiangular polygon|equiangular]] (which means that all its angles are the same) is a regular polygon. It always has the same number of edges and [[Vertex|points]].

A polygon is also [[Convex set|convex]] if there aren't any two points within it that can't be connected with a straight line, where the entire straight line is also in it. An example of a concave shape is a five-pointed [[Star polygon|star]]. It's concave because you can connect two of the points of a star with a straight line, and most of the straight line doesn't go through the star. If a five-pointed star is both equilateral and equiangular, it is also a regular polygon, but it isn't a convex one. All regular concave polygons are stars.
== Examples ==
The simplest examples of regular convex polygons are:
<gallery>
Equilateral Triangle.svg|[[equilateral triangle]]
Square diagonals.svg|[[square]]
01-Fünfeck simple.svg|[[pentagon]]
Hexagon.svg|[[hexagon]]
Heptagon.svg|[[heptagon]]
Regular octagon.svg|[[octagon]]
Nonagon.svg|[[nonagon|enneagon]]
Regular-Decagon.svg|[[decagon]]
</gallery>Polygons with more sides look more like [[circles]]. A polygon with an [[Infinity|infinite]] number of sides is called an [[apeirogon]]. Polygons can be described in text by a special symbol called a [[Schläfli symbol]]. This symbol is written as curly brackets, with some numbers inside it, like {3}. {3} is a triangle. The number inside can also be a [[Fraction (mathematics)|fraction]]. A Schläfli symbol of {5/2} is a five-pointed star.
== Properties ==
Most mathematical objects have things that can be said about them that are always true. These are called properties. All polygons (convex and star polygons) can be [[Rotation|rotated]] in place by some angle and still look the same—in other words, they're all rotationally symmetric. For example, a square could be rotated a [[Right angle|quarter turn]], and it would look the same as before. Another way of seeing this property is that any polygon can have a circle drawn around it, touching all its points, and the polygon's points are evenly spaced around the circle. A polygon can also have a circle drawn inside it, touching all its edges (but not overlapping), and the places where the circle touches the polygon are also evenly spaced. The centers of the two circles drawn are in the same place. This proves that a regular polygon is rotationally symmetric.

The circle outside the polygon is called [[Circumscribed circle|circumscribed]], and the circle inside the polygon is called [[Inscribed circle|inscribed]]. Any polygon where a circle circumscribes all its points is called a cyclic polygon. All regular polygons are cyclic, but not all cyclic polygons are regular.


== Regular Polygon Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:regular star polygons.svg|Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols

File:annuli_with_same_area_around_unit_regular_polygons.svg|As a corollary of the [[annulus (mathematics)|annulus]] chord formula, the&nbsp;area bounded by the [[circumcircle]] and [[incircle]] of every unit convex regular polygon is {{pi}}/4

File:PolygonParameters.png|Regular [[pentagon]] (''n'' = 5) with [[edge (geometry)|side]] ''s'', [[circumradius]] ''R'' and [[apothem]] ''a''

File:6-gon rhombic dissection.svg  |regular '2n'-gon dissected into rhombs from an orthogonal projection of an 'n'-cube.

File:8-gon rhombic dissection.svg  |regular '2n'-gon dissected into rhombs from an orthogonal projection of an 'n'-cube.

File:Sun_decagon.svg               |en:Decagon divided into 5 narrow and 5 wide en:Penrose rhombi

File:12-gon rhombic dissection.svg |regular '2n'-gon dissected into rhombs from an orthogonal projection of an 'n'-cube.

File:14-gon-dissection-star.svg    |en:Tetradecagon dissection by rhombi in star symmetry

File:16-gon rhombic dissection.svg |regular '2n'-gon dissected into rhombs from an orthogonal projection of an 'n'-cube.

File:18-gon-dissection-star.svg    |en:regular polygon dissection into rhombs
</gallery>
{{Polytopes}}

[[Category:Polygons]]