polygon

A polygon is a simple (http://planetmath.org/SimpleCurve), closed path that lies on a plane and is composed entirely of line segments. In other words, if someone were to walk on a polygon, then the person would end up back where he started; moreover, if a person walks on a polygon so that he travels exactly once around the polygon, then the path (http://planetmath.org/Path) never crosses itself, and the person is either walking along a line or turning.

Below are some examples of polygons:

A side of a polygon is a line segment on the polygon that is of maximal length. In other words, any line segment that contains a side of a polygon and has a greater length than that side is not entirely on that polygon. A vertex of a polygon is an endpoint of a side of the polygon. Note that each vertex of a polygon is simultaneously the endpoint of exactly two adjacent sides of the polygon.

A polygon with $n$ sides is called an $n$-gon. For small $n$, there are more traditional names:

number of sides	name of polygon
3	triangle
4	quadrilateral
5	pentagon
6	hexagon
7	heptagon
8	octagon
10	decagon

In spherical geometry, polygons with only two sides exist. They are called biangles.

The perimeter of a polygon is the sum of the lengths of its sides.

An interior angle of a polygon is the measure of an angle formed by two adjacent sides such that the angle is measured with respect to the interior of the polygon. For each polygon in the picture below, the interior angles are marked in blue:

Note that the measure (http://planetmath.org/AngleMeasure) of any interior angle of a polygon is strictly between $0^{\circ }$ and ${360}^{\circ }$ and is not equal to ${180}^{\circ }$.

We have the following criterion for a polygon to be convex:

Theorem.

A polygon is convex if and only if each of its interior angles has a measure that is strictly less than ${\mathrm{180}}^{\mathrm{\circ }}$.

The angle sum of a polygon is the sum of the measures of its interior angles. In Euclidean geometry, the angle sum of an $n$-gon is exactly $180{(n-2)}^{\circ }$.

An exterior angle of a polygon is any angle that forms a linear pair with an interior angle of a polygon. In the picture below, all exterior angles of the triangle are marked in blue:

For a more rigorous treatment of polygons, see this entry (http://planetmath.org/Polygon).

Title	polygon
Canonical name	Polygon1
Date of creation	2013-03-22 17:35:35
Last modified on	2013-03-22 17:35:35
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	20
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 51-00
Related topic	Polygon
Related topic	BasicLength
Related topic	Diagonal
Related topic	RegularPolygon
Related topic	Semiperimeter
Related topic	EquilateralPolygon
Related topic	EquiangularPolygon
Defines	side
Defines	vertex
Defines	$n$-gon
Defines	n-gon
Defines	perimeter
Defines	interior angle
Defines	angle sum
Defines	exterior angle