Polygons

Polygons are defined as two-dimensional 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, let us learn about polygons definition, regular polygons, polygon sides, and the properties of polygons, along with polygon examples and their identification.

The word 'Polygon' is derived from a Greek word in which 'poly' means 'many' and 'gon' means 'angle'. This means that a polygon is a closed figure that is formed by straight lines and these straight lines form the interior angles in it. Polygons can be commonly seen around us. For example, the shape of a honeycomb is a polygon with 6 sides and is known as a hexagon. Each polygon is different in structure and is categorized based on the number of sides and its properties. It should be noted that all polygons are closed plane shapes.

Polygon Definition

In geometry, the definition of a polygon is given as a closed two-dimensional figure which is formed by three or more straight lines.

The properties of polygons help us identify them easily. In other words, the following characteristics of a polygon help us to easily check whether a given shape is a polygon or not

• A polygon 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.
• It has three or more sides in it.
• 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.

Polygon Sides

The sides of a polygon define the name of the specific polygon because different polygons have different number of sides. For example, if a polygon has 3 sides, then it is called a triangle, whereas, if a polygon has 4 sides, it is a quadrilateral. The following section shows the different types of polygons along with their names based on the number of sides.

Types of Polygons

There are different types of polygons and they have different names depending on the number of sides that they have. For example, a 3-sided polygon is a triangle, a 4-sided polygon is a quadrilateral, a 5-sided polygon is a pentagon, a 6-sided polygon is a hexagon, and so on.

Polygon Chart

The following chart shows the naming convention of polygons on the basis of the number of sides that they have. Each polygon is given a special name on the basis of its number of sides. For example, the trigon, also known as the triangle is made of two words 'tri' which means three, and 'gon' means angles. This shows that it is a shape that has three angles. Observe the table given below to see the names of different polygons as per their number of sides.

A polygon can be categorized as a regular or irregular polygon based on the length of its sides and the measure of its angles. The difference between a regular and irregular polygon is given in the following table.

Regular Polygons and 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 is referred to as an irregular polygon. Regular polygons are always 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 has sides of different lengths and angles of different measures is referred to as an irregular polygon.

Observe the figure of a regular hexagon given below to understand the parts of a regular polygon.

Here,

• Vertices of the hexagon: A, B, C, D, E, and F
• All the sides of this regular hexagon are equal, i.e., AB = BC = CD = DE = EF = FA
• All the interior angles are equal (represented in blue)
• All the exterior angles are equal (represented in yellow)
• BE is the diagonal.

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 of the polygon

Important Notes on Polygons

• Polygons are 2-D figures with more than 3 sides.
• The 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

There are two basic formulas for polygons listed below:

• Area of polygons
• Perimeter of polygons

Let us learn about the above-listed two polygon formulas in detail.

Area of Polygons

The area of a polygon is defined as the measurement of space enclosed within a polygon. The area of polygons can be found by different formulas depending upon whether the polygon is a regular or an irregular polygon. For example, a triangle is a three-sided polygon which is known as a trigon. The formula for calculating the area of the trigon (triangle) is half the product of the base and height of the triangle. It is expressed in square units like, m², cm², ft². Similarly, each polygon has a different formula depending on the number of sides and the type of polygon.

Perimeter of Polygons

The perimeter of a polygon is defined as the distance around a polygon which can be obtained by summing up the length of all given sides.

The perimeter of polygon formula = Length of Side 1 + Length of Side 2 + Length of Side 3...+ Length of side N (for an N-sided polygon). It is expressed in terms of units such as meters, cm, feet, etc.

Concave polygons are those polygons that have at least one interior angle which is a reflex angle and it points inwards. Concave polygons have a minimum of 4 sides and a few of the diagonals in a concave polygon may lie partly or fully outside it. It is to be noted that all concave polygons are irregular because the interior angles are not equal.

On the other hand, a convex polygon has no interior angle that measures more than 180°. A convex polygon can have 3 sides and no diagonal in a convex polygon lies outside it.

Polygons Worksheets

The polygons worksheets helps children recognize more shapes and patterns in real life. It also develops the base of understanding and establishing the necessary basic background for geometry.

Download Polygons Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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What are Polygons in Math?

Polygons are closed, two-dimensional shapes that are formed by three or more line segments. They are closed, plane figures that are bounded by straight lines.

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 should be 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 Polygon and an Irregular Polygon?

A polygon in which all sides are of equal length and all angles are of equal measure is called a regular polygon. An irregular polygon is a polygon in which the sides and angles are of different lengths and measures.

☛ Also Check:

• Types of Polygon

How do you know if a Polygon is Regular?

Any polygon is a regular polygon if it satisfies the following 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 a Regular Polygon?

The angle that is formed by adjacent sides inside the polygon is referred to as the interior angle. 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 to be 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. It is not made up of straight lines and it does not have an interior or exterior angle. Therefore, it can be said that a 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 words refer to the shape of 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.

What is the Area of Polygon Shape?

The total space enclosed by a polygon in a two-dimensional plane is defined as the area of a polygon. We write the unit of area of the polygon as square units such as (meters² or centimeters², etc.) or USCS units (inches or feet, etc).

What is the Perimeter of Polygon Shape?

The perimeter of a polygon is defined as the total length of the boundary of the polygon in a two-dimensional plane. The perimeter of polygons is expressed in units like centimeters, inches, feet, and so on.

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.

What is a Regular Polygon?

A regular polygon is one in which all the sides are of equal length and all the angles are of equal measure. For example, a square is a regular polygon.

What are the Properties of a Polygon?

The properties of a polygon are listed below which help us identify a figure as a polygon:

• A polygon is a closed shape.
• It is made of line segments or straight lines.
• A polygon is a two-dimensional shape (2D shape) that has only two dimensions - length and width.
• A polygon has at least three or more sides.
• The angles and side lengths of a polygon may be of the same length or of different lengths.

How to Find the Number of Sides of a Polygon?

If the value of a single exterior angle of a regular polygon is given, then the number of sides of that regular polygon can be calculated using the formula, Number of sides of the regular polygon (n) = 360 ÷ exterior angle.

What are the Different Types of Polygons?

There are different types of polygons that are named according to the number of sides that they have. For example, a 3-sided polygon is called a triangle, a 4-sided polygon is called a quadrilateral, a 5-sided polygon is called a pentagon, a 6-sided polygon is called a hexagon, a 7-sided polygon is called a heptagon, and so on. A detailed chart of the different types of polygons is given above on this page.

What is the Difference Between Concave and Convex Polygons?

The differences between concave and convex polygons are given below.

• Concave polygons are those polygons that have at least one interior angle which is a reflex angle and it points inwards. But a convex polygon has no interior angle that measures more than 180°.
• Concave polygons have a minimum of 4 sides but a convex polygon can have 3 sides.
• A few of the diagonals in a concave polygon may lie partly or fully outside it. But no diagonal in a convex polygon lies outside it.
• It is to be noted that all concave polygons are irregular because the interior angles are not equal. However, a convex polygon may not always be an irregular polygon.