Polygon

A polygon (literally "many angle") is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary.

Names and types

Polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle and quadrilateral are exceptions. For larger numbers, mathematicians write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Naming polygons

To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows

That is, a 42-sided figure would be named as follows:

and a 50-sided figure

But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons).

Taxonomic classification

The taxonomic classification of polygons is illustrated by the following tree:

                                      Polygon
                                     /       \
                                 Simple     Complex
                                /     \
                           Convex     Concave
                            /
                       Cyclic 
                       /    
                  Regular

• A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); otherwise it is called complex.
• A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave.
• A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
• A cyclic polygon is called regular if all its sides are of equal length and all its angles are equal; for each number of sides, all regular polygons with the same number of sides are similar.

The regular polygons most commonly found include:

• equilateral triangle
• square
• regular pentagon
• regular hexagon
• regular octagon

Somewhat less common are:

• regular decagon
• regular dodecagon

See also tilings of regular polygons.

Properties

We will assume Euclidean geometry throughout.

An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape.

In the case of a line of symmetry the latter reduces to n-2.

Let k≥2. For an nk-gon with k-fold rotational symmetry (C_k), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (D_k) there are n-1 degrees of freedom.

Angles

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) turns). This can be seen in two different ways:

• Moving around a simple n-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)
• Any simple n-gon can be considered to be made up of (n−2) triangles, each of which has an angle sum of π radians or 180°.

Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics).

Area

Several formulas give the area of a regular polygon:

half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side)

The area A of a simple polygon can be computed if the cartesian coordinates (x₁, y₁), (x₂, y₂), ..., (x_n, y_n) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is

A = ½ · (x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + ... + x_ny₁ − x₁y_n)

  = ½ · (x₁(y₂ − y_n) + x₂(y₃ − y₁) + x₃(y₄ − y₂) + ... + x_n(y₁ − y_n−1))

The formula was described by Meister in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem.

If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem.

Construction

All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).

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

Point in polygon test

In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x₀,y₀) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.

Special cases

Some special cases are:

• angle of 0° or 180° (degenerate case)
• two non-adjacent sides are on the same line
• equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
• equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)

A triangle is equilateral iff it is equiangular.

An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle.

See also

• cyclic polygon
• geometric shape
• polyform
• polyhedron
• polytope
• simple polygon
• star polygon
• synthetic geometry
• tiling
• tiling puzzleda:Polygon

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