Polygon – Shape, Formula, Types, and Examples

A polygon is a two-dimensional shape made up of straight lines that form a closed polygonal chain. The word “polygon” comes from the words “poly” and “gon”, which mean “many” and “sides”. The segments of a polygon are called its sides or edges, and the points where two edges meet are called its vertices or corners. A polygon with n sides is called an n-gon. For example, a triangle is a 3-gon.

Polygons can be simple or self-intersecting. A simple polygon does not intersect itself, except at the shared endpoints of consecutive segments. A polygonal chain that crosses over itself creates a self-intersecting polygon. Polygons can also be classified as concave or convex.

In this article, we will discuss polygons, polygon definition, polygon shape, types of polygons, polygon formulas, important terminologies of Polygon, and polygon meaning. We will also solve various examples and provide practice questions for a better understanding of the concept of this article.

Important Facts about Polygons
Sum of interior angles of polygon	(n–2) × 180°
Number of Diagonals in polygon	n(n–3)/2
Interior Angle of Regular polygon	{(n–2) × 180°}/n
Exterior Angle of Regular polygon	360°/n

What are Polygons?

The term ‘Polygon’ originates from the Greek word “polugonos”, where ‘poly’ signifies ‘many,’ and ‘gon’ denotes ‘angle.’ Generally, a polygon is a closed figure formed by straight lines, with its interior angles created by these lines. To constitute a closed shape, a minimum of three-line segments is necessary. It is commonly known as a Triangle or a 3-gon. The general term for an n-sided Polygon is an n-gon.

Polygon Definition

Polygons are flat, two-dimensional figures composed of straight sides that form a fully enclosed shape. In geometry, the polygon is a plane figure made up of line segments connected to form a closed polygonal chain. They consist of straight sides, not curves, and can have any varying number of sides. Some polygons of different kinds are: open, boundary only, closed and self-intersecting.

In geometry, a Polygon is defined as a closed, two-dimensional shape that lies flat in a plane and is enclosed by straight sides.

A Polygon lacks curved sides, and its edges are the straight segments defining its boundary. The meeting points of these edges are termed vertices or corners.

Polygon Examples

In terms of mathematics triangles, hexagons, pentagons, and quadrilaterals are examples of Polygons. Real-life examples of Polygon are rectangular-shaped screen on your laptop, television, mobile phone; rectangular football pitch or playground, Bermuda Triangle and Egypt’s Pyramids of triangular shape.

Parts of a Polygon

A Polygon comprises three fundamental components:

• Sides of Polygon: Sides of a Polygons are the boundary of the polygons that define the closed region.
• Vertices: The point at which two sides meet is known as a vertex.
• Angles: The Polygon contains both interior and exterior angles. An interior angle is formed within the enclosed region of the polygon by the intersection of its sides.

Polygon Chart based on Number of Sides

Nomenclature of Polygon defined on the basis of number of sides they possess. It is designated as n-gons, where ‘n’ signifies the number of sides. Polygons are generally identified by the quantity of their edges. For example, a polygon with five sides is termed a 5-gon, while one with ten sides is referred to as a 10-gon.

Polygon Chart
Polygon Shape Names	Number of Sides	Number of vertices	Number of diagonals	Interior Angle Measure
Triangle	Polygons with 3 sides	3	0	60°
Quadrilateral	Polygons with 4 sides	4	2	90°
Pentagon	Polygons with 5 sides	5	5	108°
Hexagon	Polygons with 6 sides	6	9	120°
Heptagon	Polygons with 7 sides	7	14	128.571°
Octagon	Polygons with 8 sides	8	20	135°
Nonagon	Polygons with 9 sides	9	27	140°
Decagon	Polygons with 10 sides	10	35	144°
Hendecagon	Polygons with 11 sides	11	44	147.273°
Dodecagon	Polygons with 12 sides	12	54	150°

Properties of Polygons

The properties of Polygons identify them easily. The following properties contribute to know the Polygons easily:

• A polygon is a closed shape, devoid of open ends. Origin and end point should be same.
• It assumes a planar form, consisting of line segments or straight lines that collectively shape the figure.
• As a two-dimensional entity, a polygon exists only in the dimensions of length and width, lacking depth or height.
• It possesses three or more sides to make a polygon.
• Angles in the Polygon can vary. It shows distinct configuration.
• Length of the sides of a Polygon can vary; it may or may not be the equal across the Polygon.

Polygon Shapes

A Polygon is a flat, two-dimensional shape characterized by straight sides connected to form a closed figure. Examples of Polygon shapes include:

• Triangle
• Quadrilateral
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon

Triangle

• It has 3 sides and 3 vertices.
• It has no diagonals.
• Sum of the interior is 180°.

Quadrilateral

• It has 4 sides and 4 vertices.
• It has 2 diagonals.
• Sum of the interior angle is 360°.

Pentagon

• It has 5 sides and 5 vertices.
• It has 5 diagonals.
• Sum of the interior angle is 540°.

Hexagon

• It has 6 sides and 6 vertices.
• It has 9 diagonals.
• Sum of the interior angle is 720°.

Heptagon

• It has 7 sides and 7 vertices.
• It has 14 diagonals.
• Sum of the interior angle is 900°.

Octagon

• It has 8 sides and 8 vertices.
• It has 20 diagonals.
• Sum of the interior angle is 1080°.

Nonagon

• It has 9 sides and 9 vertices.
• It has 27 diagonals.
• Sum of the interior angle is 1260°.

Decagon

• It has 10 sides and 10 vertices.
• It has 35 diagonals.
• Sum of the interior angle is 1440°.

Types of Polygons

Depending on the sides and angles, the Polygons can be classified into different types:

Polygons on the Basis of Sides

Polygons can be categorized based on the characteristics of their sides into two primary types:

• Regular Polygon
• Irregular Polygon

Regular Polygon

A Regular Polygon is distinguished by having all sides of equal length and all interior angles with equal measurements. It can be both equilateral and equiangular. Examples of regular polygons include the triangle, quadrilateral, pentagon, and hexagon.

Regular Polygon

Irregular Polygon

An Irregular Polygon has unequal length sides and angles of varying measures. Any polygon that does not conform to the criteria of a regular polygon is classified as irregular. Common examples of irregular polygon are the scalene triangle, quadrilaterals like rectangle, trapezoid, or kite, as well as irregular pentagon and hexagon structures.

Irregular Polygon

Polygons On Basis of Angles

Polygons can be classified based on the nature of their angles into two main categories:

• Convex Polygon
• Concave Polygon

Convex Polygon

A convex polygon has no interior angle that measures more than 180°. Convex polygons can have three or more sides. In convex polygons, all diagonals lie inside the closed figure. Common examples of convex polygons are triangles, all convex quadrilaterals, as well as regular pentagons and hexagons

Concave Polygon

A concave polygon has at least one interior angle that is a reflex angle and points inwards. Concave polygons have a minimum of four sides. This type of polygon features at least one interior angle measuring more than 180°. In concave polygons, some diagonals extend outside the enclosed figure. Examples of concave polygons include a dart or an arrowhead in quadrilaterals, as well as certain irregular pentagons and hexagons.

Concave vs Convex Polygons

Let’s see the difference between Convex and Concave Polygon in the table below:

Convex Polygon	Concave Polygon
The entire perimeter of a convex shape extends outward without any inward indentations.	A Concave shape features at least one inward-pointing portion, indicating the presence of a dent.
In a Convex Polygon, all internal angles are below 180°.	In a Concave Polygon, there exists at least one interior angle exceeding 180°.
Any line connecting two vertices of a convex shape lies entirely within the boundaries of the shape.	The line connecting any two vertices of a concave shape may or may not intersect the interior of the shape.

Polygons On Basis of Boundary

Polygons can be categorized based on the nature of their boundaries into two primary types:

• Simple Polygon
• Complex Polygon

Simple Polygon

A Simple Polygon is characterized by a singular, non-intersecting boundary. In other words, it does not cross itself, and it consists of one boundary.

Simple Polygons

Complex Polygon

On the other hand, a Complex Polygon is defined by intersect itself. It consists of more than one boundary within its structure. In Complex polygons boundary intersects, creating multiple distinct regions within the polygon.

Complex Polygon

Polygon Formulas

All the formulas related to polygons are discussed below:

Area of Polygons

Area of a Polygon represents the total space it occupies in a two-dimensional plane, is determined by specific formulas based on the number of sides and the polygon’s classification. The area formulas are as follows:

Area of Polygon Formula
Area of Polygon	Formula
Area of Triangle	1/2 × Base × Height
Area of the Parallelogram	Base × Height
Area of a Rectangle	Length × Breadth
Area of Square	(Side)2
Area of a Rhombus	1/2 × diagonal1 × diagonal2
Area of Trapezium	1/2 × Height × Sum of Parallel Sides
Area of Pentagon	(5/2) × side length × Apothem
Area of Hexagon	{(3√3)/2}side2
Area of Heptagon	3.643 × Side2

Perimeter of Polygons

The Perimeter of a two-dimensional shape represents the total length of its outer boundary. For Polygons, the Perimeter is calculated as follows:

Perimeter of Polygons Formula
Perimeter of Polygon	Formula
Perimeter of Triangle	Sum of Three Sides
Perimeter of Parallelogram	2(Sum of Adjacent Sides)
Perimeter of Rectangle	2(length + breadth)
Perimeter of Square	4 × Side
Perimeter of Rhombus	4 × Side
Perimeter of Trapezium	Sum of Parallel Sides + Sum of Non-Parallel Sides
Perimeter of Pentagon	5 × Side
Perimeter of Hexagon	6 × Side
Perimeter of Heptagon	7 × Side

Interior Angle Formula of Polygons

The Interior Angles of a Polygon are those formed between its adjacent sides and are equal in the case of a regular polygon. The count of interior angles corresponds to the number of sides in the polygon.

The sum of the interior angles ‘S’ in a polygon with ‘n’ sides is calculated as

S = (n – 2) × 180°

where ‘n’ represents the number of sides.

Exterior Angle Formula of Polygons

Each Exterior Angle of a Regular Polygon is formed by extending one of its sides (either clockwise or anticlockwise) and measuring the angle between this extension and the adjacent side. In a regular polygon, all exterior angles are equal

Total sum of exterior angles in any polygon is fixed at 360°

Therefore,

Each exterior angle is given by 360°/n

where ‘n’ is the number of sides.

Sum of the interior and corresponding exterior angles at any vertex in a polygon is always 180 degrees, expressing a supplementary relationship:

Interior angle + Exterior angle = 180 degrees

Exterior angle = 180 degrees – Interior angle

Diagonal of Polygon Formula

A Diagonal of a Polygon is a line segment formed by connecting two vertices that are not adjacent.

Number of Diagonals in a Polygon = n(n−3)/2,

where ‘n’ represents the number of sides the Polygon possesses.

Polygon – Conclusion

• Polygon is a closed figure bounded by three or more line segments
• Sum of Interior Angles: The sum of all interior angles in an n-sided polygon is given by the formula (n–2)×180°.
• Number of Diagonals: For a polygon with n sides, the number of diagonals is calculated using the formula n(n–3)/2.
• Triangles Formed by Diagonals: The number of triangles formed by joining diagonals from a single corner of a polygon is n–2.
• Interior Angle of Regular Polygon: The measure of each interior angle in an n-sided regular polygon is {(n–2)×180°}/n.
• Exterior Angle of Regular Polygon: The measure of each exterior angle in an n-sided regular polygon is 360°/n.

Polygon – Solved Examples

Example 1: Consider a quadrilateral with four sides. Find the sum of all its interior angles of quadrilateral.

Solution:

Formula for the sum of interior angles in an n-sided regular polygon = (n − 2) × 180°

The sum of all the interior angles of the quadrilateral = (4 – 2) × 180°

The sum of all the interior angles of the quadrilateral = 2 × 180°

The sum of all the interior angles of the quadrilateral = 360°

Therefore, the sum of all the interior angles of the quadrilateral is 360°.

Example 2: Consider a Regular Polygon with a given exterior and interior angle ratio of 7:3. Determine the type of polygon.

Solution:

The ratio of the exterior and interior angle is 7:3.

Assume the exterior and interior angle of a polygon as 7x and 3x.

The sum of the exterior and interior angles of any polygon is 180°.

7x + 3x = 180°

10x = 180°

x = 18°

Exterior angle = 18°

Number of sides = 360°/exterior angle

= 360°/18°

= 20

Therefore, the given polygon is an icosagon, as it has 20 sides.

Example 3: Each Exterior Angle of a Polygon measures 90 degrees, determine the type of Polygon?

Solution:

As per the formula, each exterior angle = 360°/n

Here n=number sides.

90°= 360°/n

n = 360°/90°= 4

Hence, the Polygon in question is a quadrilateral, as it possesses four sides.

Example 4: The sides are 10m, 10m, 8m, 8m, 5m, 5m, 9m, 9m. How many meters of rope will be needed for the Perimeter?

Solution:

In order to find the length of the rope needed for the perimeter, we must sum the lengths of all the sides:

Perimeter = 10 m + 10 m + 8 m + 8 m + 5 m + 5 m + 9 m + 9 m

Perimeter = 64 m.

Therefore, a total of 64 meters of rope will be needed for the Perimeter.

Polygon – Practice Questions

Following are some practice questions based on the formula of polygons:

Q1. Given one angle of a pentagon is 140°, determine the size of the largest angle if the remaining angles are in the ratio 1:2:3:4.

Q2. If the sum of the interior angles of a polygon is 160°, find the number of sides in the Polygon.

Q3. The number of sides in two regular polygons are in the ratio 2:3, and the ratio of their interior angles is 4:5. Find the respective numbers of sides of these Polygons.

Q4. Determine the total sum of angles in a heptagon.

Q5. Calculate the sum of exterior angles in a pentagon.

Practice Questions on Polygons

Answer the following MCQs (Here only one answer is correct)

1. How many sides does a hexagon have?

1. 4
2. 6
3. 8
4. 10

2. Which of the following is not a regular polygon?

1. Triangle
2. Square
3. Pentagon
4. Parallelogram

Polygons – Frequently Asked Questions

What is a Polygon in Geometry?

In mathematics, a Polygon refers to a closed two-dimensional figure formed by the connection of three or more straight lines. The term “polygon” is derived from the Greek language, with “poly-” signifying “many” and “gon” representing “angle.”

Which is Called Polygon?

A two-dimensional closed figure made up of line segments with no curves is called polygons.

Which is the Smallest Polygon?

Smallest polygon formed is triangle with three sides.

What is 20-gon?

A 20-gon is twenty-sided polygon in geometry.

What is the Total Sum of External Angles of Polygon?

Sum of the exterior angles of a Polygon is 360°.

Can a Circle be Classified as a Polygon?

Polygon is a closed shape made up of straight-line segments. The circle is a closed figure, but it is made of a curve. So, a circle is not a polygon.

What is the Sum of Interior Angle of a Polygon?

Sum of interior angle of a polygon is given by (n–2)×180° where, n is number of sides in the polygon.