Polygon

For other uses, see Polygon (disambiguation).

In geometry a polygon (IPA: [ˈpɒliɡən], from the Classical Greek πολυγον, meaning literally "many-knee" or "many-angle") is a plane figure that is bounded by a closed path or circuit, composed of a finite number of sequential line segments. The straight line segments that make up the boundary of the polygon are called its edges or sides and the points where the edges meet are the polygon's vertices or corners. The interior of the polygon is its body.

The idea of a polygon has been generalised in various ways, for example:

• A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polyhedra are classic examples.
• A spherical polygon is a circuit of sides and corners on the surface of a sphere.
• An apeirogon is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.

A polygon is a 2-dimensional example of the more general polytope in any number of dimension.

Names and types

A simple concave hexagon

A complex pentagon

Polygons are named and classified 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 large numbers, mathematicians usually 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.

Some special polygons also have their own names, for example the regular star pentagon is also known as the pentagram.

Polygon names

Name	Edges
henagon (or monogon)	1
digon	2
triangle (or trigon)	3
quadrilateral (or tetragon)	4
pentagon	5
hexagon	6
heptagon (avoid "septagon" = Latin [sept-] + Greek)	7
octagon	8
enneagon (or nonagon)	9
decagon	10
hendecagon (avoid "undecagon" = Latin [un-] + Greek)	11
dodecagon (avoid "duodecagon" = Latin [duo-] + Greek)	12
tridecagon (or triskaidecagon)	13
tetradecagon (or tetrakaidecagon)	14
pentadecagon (or quindecagon or pentakaidecagon)	15
hexadecagon (or hexakaidecagon)	16
heptadecagon (or heptakaidecagon)	17
octadecagon (or octakaidecagon)	18
enneadecagon (or enneakaidecagon or nonadecagon)	19
icosagon	20
No established English name ("hectagon" is bad Greek, "centagon" is a Latin-Greek hybrid; neither is widely attested.	100
chiliagon	1000
myriagon	10,000
googolgon	10100

Naming polygons

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

Tens		and	Ones		final suffix
		-kai-	1	-hena-	-gon
20	icosi-		2	-di-
30	triaconta-		3	-tri-
40	tetraconta-		4	-tetra-
50	pentaconta-		5	-penta-
60	hexaconta-		6	-hexa-
70	heptaconta-		7	-hepta-
80	octaconta-		8	-octa-
90	enneaconta-		9	-ennea-

In many cases KAI is not necessary.

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

Tens	and	Ones	final suffix	full polygon name
tetraconta-	-kai-	-di-	-gon	tetracontakaidigon

and a 50-sided figure

Tens	and	Ones	final suffix	full polygon name
pentaconta-			-gon	pentacontagon

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

Taxonomic classification

There is no full agreement on the taxonomy of polygons. Here is one approach.

A polygon is called simple if its boundary is described by exactly one closed path that has no self-intersections. If several closed and non-intersecting paths are needed to describe its boundary, the polygon has holes and is not simple.

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 sometimes called complex (though this has other meanings).
• 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 polygon is called regular if it is both cyclic and equilateral; all convex regular polygons with the same number of sides are similar to each other.

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 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).

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:

$A={\frac {n}{4}}s^{2}\cot {\cfrac {\pi }{n}}$

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° (digon) or 180° (apeirogon) (degenerate cases)
• Two non-adjacent sides are on the same line.
• Equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
• Isogonal or vertex-transitive polygon: an equiangular polygon whose vertices lie within the same symmetry orbit.
• Equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)
• Isotoxal or edge-transitive polygon: an equilateral polygon whose sides lie within the same symmetry orbit.

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.

External links

• Polygon name generator: type in the number of sides and see the polygon's name!
• Weisstein, Eric W. "Polygon". MathWorld.
• What Are Polyhedra? (with Greek Numerical Prefixes)
• Polygons, types of polygons, and polygon properties With interactive animation
• How to draw monochrome orthogonal polygons on screens, by Herbert Glarner

See also

• Constructible polygon
• 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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