Polygons

Polygons are defined as a plane (two-dimensional) and closed shapes that are formed by joining three or more line segments with each other. We tend to encounter polygons mostly while we learn about geometry. In this lesson, we will learn about polygons and their identification using charts in detail.

In geometry, the definition of a polygon is given as a closed two-dimensional figure with three or more straight lines. The Greek word "Polygon" consists of Poly meaning "many" and gon meaning "angle". We see many different polygons around us. For example, the shape of a honeycomb is a hexagon. Each polygon is different in structure, they are categorized based on the number of sides and their properties. Thus, all polygons are closed plane shapes.

We can identify a polygon by checking the following characteristics in a shape:

• It is a closed shape, that is, there is no end that is left open in the shape. It ends and begins at the same point.
• It is a plane shape, that is, the shape is made of line segments or straight lines.
• It is a two-dimensional figure, that is, it has only two dimensions length and width. There is no depth or height to it. 
• The shape must have three or more sides.
• The angles in the polygon may or may not be the same. 
• The length of the sides of a polygon may or may not be the same.

In order to understand polygons and their naming convention see the chart given below.

Polygon Chart

This chart shows the naming convention of polygons on the basis of the number of their sides. Each polygon is given a special name on the basis of its number of sides, such a way that when the name of the polygon is written its one part is also influenced by the number of its sides. For example, trigon, also known as the triangle is made of two words "tri" which means three, and gon mean angles which refers that it is a shape having three angles.

A polygon can be categorized as regular and irregular polygon based on the length of its sides. As the name suggests "regular" in regular polygon literally means a definite pattern that appears in the regular polygon while on the other hand "irregular" in irregular polygon means there is an irregularity that appears in a polygon. Let us learn about them individually. 

The difference between regular and irregular polygon is given as:

Regular vs Irregular Polygons

It is said that as per Euclidean Geometry, a polygon that is equiangular and equilateral is called a regular polygon while a polygon whose sides are not equiangular and equilateral referred to as an irregular polygon. A regular polygon is always prefixed by the term "regular". Regular polygons are convex i.e., all the interior angles measure less than 180º.

In simple words, a regular polygon has all angles of the same measure at each vertex and all sides of the same length while a polygon that does not have sides of the same length and measurement of angles at each vertex different is referred to as an irregular polygon.

Considering the below figure of a regular hexagon, let us discuss the parts of a regular polygon:

• Vertices
• Sides
• Interior Angles
• Exterior Angles
• Diagonals

Where,

• Vertices are A, B, C, D, E, and F
• Equal sides are AB, BC, CD, DE, EF, FA
• BE is the diagonal.
• All the interior angles are equal (represented by blue color in the figure)
• All the exterior angles are equal (represented by yellow color in the figure)

As we learned above, there are two kinds of angles that can be found in the case of a regular polygon. They are:

• Interior Angles of a Polygon
• Exterior Angles of a Polygon

Interior Angles of a Polygon

The interior angles are formed between the adjacent sides inside the polygon and are equal to each other in the case of a regular polygon. The number of interior angles is equal to the number of sides. The value of an interior angle of a regular polygon can be calculated if the number of sides of the regular polygon is known by using the following formula:

Interior angle = 180º(n-2)/n, where n is the number of sides

Exterior Angles of a Polygon

Each exterior angle of a regular polygon is formed by extending one side of the polygon (either clockwise or anticlockwise) and then the angle between that extension and the adjacent side is measured. Each exterior angle of a regular polygon is equal and the sum of the exterior angles of a polygon is 360°. An exterior angle can be calculated if the number of sides of a regular polygon is known by using the following formula:

Exterior Angle = 360º/n, where n is the number of sides

Important Notes

• Polygons are 2-D figures with more than 3 sides.
• Angles of a regular polygon can be measured by using the following formulas:
  Exterior Angle = 360º/n
  Interior angle = 180º(n-2)/n, where n refers to the number of sides.
• The sum of interior and exterior angles at a point is always 180º as they form a linear pair of angles
• For an 'n'-sided polygon, the number of diagonals can be calculated with this formula, n(n-3)/2.

Topics Related to Polygons

Check out these interesting articles to know more about polygon and its related topics.

• Polygons Worksheets
• Triangles
• Quadrilateral
• Pentagon Shape
• Hexagon

What are Polygons in Math?

The plane closed shapes which comprise three or more line segments are referred to as polygons. The world polygon as the name suggests is made of two words "poly" and "gon" where the word poly means "many" and the gon means "angles". Polygons are always two-dimensional in shape.

How to Identify a Polygon?

A shape is a polygon if it has the following characteristics:

• The shape must be a closed shape, that is, it must end and begin at the same point.
• The shape is a plane shape, that is, the shape is made of line segments or straight lines.
• The shape must be a two-dimensional figure, that is, it must have only two dimensions length and width.
• It must have three or more sides.
• The angles in the polygon may or may not be the same. 
• The length of the side of a polygon may or may not be the same.

What is the Difference Between a Regular and Irregular Polygon?

A polygon whose length of all sides is equal with equal angles at each vertex is called a regular polygon, while an irregular polygon is a polygon whose sides are not equal and angles differ from each other. These are the parameters that help us in differentiating between a regular and an irregular polygon.

How do you know if a Polygon is Regular?

Any polygon is a regular polygon if it satisfies the below three criteria:

• The length of all its sides must be equal.
• All interior angles must measure the same.
• All the exterior angles must measure the same.

What is the Interior Angle of Regular Polygon?

The angle that is formed by adjacent sides inside the polygon is referred to as the interior angle. The values of all the interior angles in a regular polygon are equal to each other. The value of an interior angle of a regular polygon can be calculated by using the following formula, interior angle = 180º(n-2)/n, where n is the number of sides.

Is a Circle Considered a Polygon?

No, a circle is not considered a polygon because it is not made up of three or more straight lines or line segments. It does not fulfill the criterion which can be used to identify a polygon, as it neither has three or more sides nor it shows any angles. Thus, we say the circle is not a polygon.

What is an 11 Sided Polygon Called?

An 11 sided polygon is referred to as Hendecagon. It is derived from two Greek words "Hendeka" which means eleven and "gon" which means angles. Both the words cumulatively refer to the shape being an eleven-sided polygon.

How are Polygons Named?

The name of each polygon is made of two words. The first part of the word is influenced by the Greek meaning of the number of sides it possesses and it is suffixed by the word gon. For example, the word "Hexagon" is made of two words "hex" and "gon". The word "hex" means the number six and "gon" means the angles. The words "Triangle" and "Quadrilateral" stand as an exception in this case as they are not given as per the nomenclature.

Are Polygons Always Closed Shapes?

Yes, polygons are always closed shapes as they are made of three or more straight lines which begin and start at the same point. For any shape to be a polygon it is necessary that it is a closed shape. This is also one of the most important criteria used to identify a shape as a polygon.

Are All Triangles Polygons?

Yes, all triangles are polygons as they follow all the criteria of being a polygon. In the case of any triangle, whether equilateral triangle, isosceles triangle, or scalene triangle, the following criteria are always satisfied:

• The shape ends and begins at the same point.
• It is made of line segments or straight lines. 
• It has only two dimensions that are length and width.
• It has three sides.
• The angles may or may not be the same. 
• The length of the side of a polygon may or may not be the same. 

Thus the above-given points prove that all triangles are polygons.