Polygon

A polygon is a two-dimensional geometric closed figure bounded by a continuous set of line segments. A polygon, in Euclidian geometry, must have at least three sides. A polygon of three sides is called a triangle, four sides a quadrilateral, five sides a pentagon, six sides a hexagon. Figures with more sides are typically named with the Greek name for the number of sides, followed by "-gon". Mathematicians discussing the properties of polygons with large numbers of sides will often use the formulation n-gon, where n is replaced by the number of sides (ie, a 17-gon or a 100-gon). When discussing the properties of classes of polygons which include polygons of different numbers of sides, mathematicians will sometimes refer to n-gon, without substituting the n.

The line segments bounding a polygon are known as sides, and the points where the sides meet are vertices (singular vertex).

A polygon is known as a simple polygon if none of its sides cross other sides, and each vertex is the meeting point of only two sides. A polygon which does not meet this criterion is a complex polygon. A polygon is called convex when there are no line segments which connect two points within the polygon which pass outside the polygon. Convex polygons have no internal angles between two adjacent sides greater than 180 degrees of arc. A polygon which has all sides of equal length is known as an equilateral polygon. A polygon which has all internal angles equal is known as an equiangular polygon. If the number of sides is greater than three, an equilateral polygon is not necessarily an equilateral polygon. A convex polygon which has all sides and all internal angles equal is known as a regular polygon. A complex polygon which has all sides and all angles equal is known as a regular star polygon.

Properties of polygons

All polygons have the same number of sides and vertices. The sum of the interior angles of a simple polygon, R, is $R=\pi *(n-2)$, or $R=180*(n-2)$ (in degrees) where n is the number of sides of the polygon. Other than for triangles and quadrilaterals, there are no general formulas for determining the area of a polygon; the area must be determined by dividing the polygon into separate pieces whose areas can be determined (usually triangles), and adding the area of all the parts.

Regular polygons have properties which are more easily determined analytically. For a given regular polygon with side length s and number of sides n, interior angle at each vertex is $\theta ={\tfrac {\pi (n-2)}{n}}=\left(1-{\tfrac {2}{n}}\right)\pi$, the perimeter p is: $p=sn$, and the area A is: $A={\tfrac {1}{4}}s^{2}\cot \left({\tfrac {\pi }{n}}\right)$ .

Named polygons

Many polygons are named, and for 3-gons and 4-gons, there are particular names for special cases. Some of these are listed below.

number of sides	name	properties
3	triangle
	right triangle	One internal angle is a right angle (90 degrees)
	isoceles triangle	Two sides of the same length
	equilateral triangle	All three sides of the same length, three angles equal (60 degrees).
4	quadrilateral, quadrangle, tetragon	quadrangle and tetragon are not common usages
	trapezoid	two sides parallel
	parallelogram	both pairs of non-adjacent sides are parallel
	rhombus	equilateral (also a parallelogram)
	rectangle	equiangular (also a parallelogram)
	square	regular
5	pentagon
6	hexagon
7	heptagon
8	octagon
9	nonagon
10	decagon
n>10	usually n-gon	there are rules for using Greek numbers or constructing polygon names, but these are not frequently used.