# Polygon

A polygon is a two-dimensional geometric closed figure bounded by a continuous set of line segments. The word derives from the Greek word for angle, γωνία, and the Greek word for many, πολλος (also πολυς), hence literally a polygon is a "many-angle". A polygon, in Euclidean 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 numeral 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 (i.e., 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 equiangular 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 = π * (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).
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.