Polygons are closed figures formed by line segments. The closed nature of polygons lead to geometric measures like area and perimeter. Polygons are named based on the number of segments enclosing them and classified on the basis of equality in the measures of these line segments and how the vertices are pointing toward. Polygons can also be partitioned into smaller polygons in any manner as desired.

## Definition of Polygon

A polygon is a closed shape formed by a finite number of non collinear line segments all lying on the same plane. The line segments forming the polygon are called the sides of Polygon and each side intersects exactly two other sides only at their end points.
A polygon is a closed figure formed only by line segments. The sides of each angle in a polygon are called the sides of the polygon, and the vertex of each angle of the polygon is also known as a vertex of the polygon. A Polygon is named after the letters corresponding to the vertices in consecutive order.

In the above diagram a six sided polygon known as a Hexagon is shown. The vertices of the Polygon ABCDEF are A, B, C, D, E and F. The sides of the polygon are AB, BC, CD, DE, EF and FA.

In general a polygon with n sides also have n vertices and n angles.

## Polygon Shapes

Polygons are named after the number of sides enclosing them. Some of the commonly known shapes are shown here.

 A triangle is a three sided polygon.Triangle is the polygon with the minimumnumber of sides, as at least three sidesare required to enclose area, A quadrilateral is a polygon with four sides.Special quadrilaterals like Parallelogram, Rectangle, Square, Rhombus, Trapezoidand Kite are also four sided Polygons. A Pentagon is a five sided polygon.
Similarly, Hexagon, Heptagon and Octagon are six, seven and eight sided polygons respectively.

A general Polygon with n sides is called a n-gon.

## Types of Polygons

Polygons are classified according to the measure of the interior angles as Convex and Concave Polygons.
If the measures of all the interior angles of a Polygon are less than 180º, then it is called a Convex Polygon.In this case all the vertices of the Polygon will be pointing outward. Also the lines containing each side of the Polygon will contain points in the interior of the Polygon.

If the measure of at least one interior angle of a Polygon is greater than 180º, then it is called a Concave Polygon.
In this case some of the vertices of the Polygon will be pointing inward and the lines containing some sides of the Polygon will contain points in the interior of the Polygon.

Convex Polygons are in turn classified as Regular and irregular polygons While the Concave Polygons are always irregular.

## Regular Polygons

Convex Polygon in which all the sides are congruent and all the angles are congruent is called a regular Polygon.

A regular polygon is both equiangular and equilateral. An equilateral triangle and a square are examples of regular Polygons. In a regular polygon all the vertex angles are also congruent.

Measure of each interior angle of a regular polygon of n sides= $\frac{(n-2)180}{n}$ degrees.

Measure of each exterior angle of a regular polygon of n sides = $\frac{360}{n}$ degrees.

Measure of each vertex angle = $\frac{360}{n}$ degrees.

## Irregular Polygons

Polygons in which not all sides are congruent and not all angles are congruent is called a irregular Polygon. An irregular Polygon can be either Convex or Concave. All triangles but equilateral triangles are irregular polygons. All parallelograms other than squares are irregular polygons.

An irregular polygon can either be equiangular or equilateral but never can be both.
A rectangle is equiangular but not equilateral. Hence it is an irregular Polygon.
A rhombus is an irregular Polygon, even though it is equilateral, as it is not equiangular.
Sum of the measures of the interior angles of a convex irregular polygon of n sides = (n - 2)180 degrees.

## Properties of Polygons

Regular polygons have following properties
1. All sides are congruent
2. All angles are congruent. Measure of each interior angle = $\frac{(n-2)180}{n}$ degrees where n is the number of sides of the polygon.
3. The measure of an exterior angle of a regular n-gon is $\frac{360}{n}$ degrees.
4. Area of a regular polygon = $\frac{1}{2}$ x Apothem x Perimeter
5. A regular polygon can be circumscribed by a circle.
6. A regular polygon can inscribe a circle.

There are other special properties associated to special quadrilaterals like Parallelograms, Rectangles, Kites and Trapeziums which are irregular polygons.

## Examples of Polygons

Examples of Polygons stating the type to which they belong.
 Four vertices are pointing inward.The Polygon has eight congruent sidesHence it is a Concave Octagon.Being a Concave polygon this cannot be aregular polygon. Even though the sides are congruent, the interior angles are notcongruent. Hence this is an irregular Polygon.It can be called as a Concave equilateralOctagon. Five sided Convex Polygon.All the five sides are congruent., butthe angels are not. Hence this is nota regular polygon.This can be called an equilateralirregular Pentagon. A convex Polygon in which all the ten sidesare congruent and all the ten angles are congruent. The Polygon is both equilateral and equiangular and hence regular.This is a regular decagon.