Polygon

From Wikipedia, the free encyclopedia

For other uses, see Polygon (disambiguation).

'Polygon' (IPA: [ˈpɒliɡən], from Greek, 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 edges or sides and the points where the edges meet are the polygon's vertices. If a polygon is simple, then its edges (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.

[edit] 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, quadrilateral, and nonagon are exceptions. For larger numbers, some 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.

[edit] Naming polygons

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

In many cases KAI is not necessary.

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

and a 50-sided figure

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

[edit] Taxonomic classification

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

                                      Polygon
                                     /       \
                                 Simple     Complex
                                /      \      /
                           Convex    Concave /
                            /    \     /    /
                       Cyclic    Equilateral
                           \     /    
                           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 or non-convex.
• A polygon is called equilateral if all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex or even simple.)
• A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
• A simple polygon is called regular if it is both cyclic and equilateral; all regular polygons with the same number of sides are similar to each other.

[edit] 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.

[edit] Angles

Any polygon, regular or irregular, complex or simple, has as many angles as it has sides.

• The sum of the interior angles of a simple n-gon is (n−2)π radians or (n−2)180 degrees. This is because 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 degrees. In topology and analysis, the sum of the interior angles can be seen as 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 -½ and ½ winding.)

• The measure of any interior angle of a regular n-gon is (n−2)π/n radians or (n−2)180/n degrees.

• Exterior angles are formed by extending a side beyond the polygon. The measure of the exterior angle added to the measure of the adjacent interior angle is 180 degrees.

• 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).

[edit] Area

The area of a polygon is the measurment of the 2-dimensional region enclosed by the polygon.

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.

For a regular polygon with n sides of length s, the area is given by:

<math>A = \frac{n}{4} s^2 \cot{\cfrac{\pi}{n}} </math>

In simple terms a polygon is a multi sided shape made of verticies, edges and faces.

[edit] 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.

[edit] 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 if and only if it is equiangular.

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

A quadrilateral is a square if and only if it is both equilateral and equiangular. Likewise, a quadrilateral is a square if and only if it is both a rhombus and a rectangle.

[edit] External links

• Polygon name generator: type in the number of sides and see the polygon's name!
• Mathworld: Polygon
• [1] More information
• What Are Polyhedra? (with Greek Numerical Prefixes)
• Polygons, types of polygons, and polygon properties With interactive animation
• Interior angles of polygons With interactive animation
• Exterior angles of polygons With interactive animation. Includes treatement of exterior angles when the polygon is concave.
• Exterior / Exterior angle relationship of a polygon With interactive animation. Shows they are supplementary even if the vertex is concave.
• C code to calculate the area of a simple (non-self-crossing) polygon

[edit] See also

• Cyclic polygon
• Geometric shape
• Polygon triangulation
• Polyform
• Polyhedron
• Polytope
• Regular polygon
• Simple polygon
• Star polygon
• Synthetic geometry
• Tiling
• Tiling puzzle
• Golygon

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