Polygon: Definition, Shape, Area Formula, Properties, Examples

What is a Polygon?

A polygon is one of the most important concepts in geometry and forms the foundation for studying two-dimensional shapes. From simple triangles and rectangles to complex geometric figures, polygons are everywhere in mathematics and daily life. Understanding polygons helps students learn about sides, angles, diagonals, perimeter, area, and other geometric properties that are frequently used in school mathematics and competitive examinations.

Polygon Meaning in Simple Words

In simple words, a polygon is a closed shape formed by joining three or more straight line segments.

A polygon must:

• Be a closed figure
• Have only straight sides
• Have sides that meet only at their endpoints

Examples of Polygons

• Triangle
• Square
• Rectangle
• Pentagon
• Hexagon
• Octagon

Non-Examples of Polygons

• Circle (contains a curved boundary)
• Semicircle
• Open figures
• Shapes with curved sides

For example, a triangle is a polygon because it is a closed figure made of three straight sides, whereas a circle is not a polygon because it contains no straight sides.

Definition of Polygon

A polygon is a closed two-dimensional geometric figure formed by a finite number of straight line segments connected end to end.

The word "polygon" comes from Greek words:

• "Poly" meaning many
• "Gon" meaning angles

Therefore, polygon means "many-angled figure."

Characteristics of a Polygon

• It is a plane (2D) figure.
• It has three or more sides.
• All sides are straight line segments.
• It is completely closed.
• Adjacent sides meet at vertices.

Example

A pentagon has:

• 5 sides
• 5 vertices
• 5 interior angles

Hence, it is a polygon.

Real-Life Examples of Polygons

Polygons can be seen all around us in everyday objects, buildings, signs, and designs. Understanding real-life polygon examples helps connect geometry with practical applications.

Common Real-Life Polygon Examples

• A rectangular television screen.
• A triangular road warning sign.
• A square chessboard.
• A hexagonal floor tile.
• An octagonal stop sign.
• A pentagonal building design.

These examples demonstrate how polygons are used in architecture, engineering, construction, graphic design, and everyday objects.

Why Polygons are Important in Geometry

Polygons play a fundamental role in geometry because many geometric concepts are based on them. Topics such as angles, area, perimeter, symmetry, and coordinate geometry rely heavily on polygon properties.

Importance of Polygons in Mathematics

• Help understand geometric shapes and structures.
• Form the basis of area and perimeter calculations.
• Used in studying interior and exterior angles.
• Essential for coordinate geometry.
• Important in trigonometry and mensuration.
• Used in geometric proofs and constructions.
• Help develop spatial reasoning skills.

Applications of Polygons

Polygons can be regular or irregular.

Triangles, squares, and rectangles are some of the examples of polygon.

Different shapes of a Polygon

Polygons have different shapes depending on the number of sides they have.

Some of the very common shapes are:

Triangles like equilateral triangles, isosceles triangles, and scalene triangles.

Quadrilaterals like squares, rectangles, rhombuses, parallelograms and kites.

Some of the pictures of different shapes are given below.

List of Polygons of sides up to 12

Types of polygon

Polygons can be categorised based on their sides and angles into several types. Here are the main types of polygons:

• Regular Polygon

• Irregular Polygon

• Convex Polygon

• Concave Polygon

Now we will discuss these types of polygons thoroughly.

Regular polygons

A polygon with equal sides and equal angles is called a Regular polygon.

Key properties of regular polygons are:

• All interior angles are equal.

• All sides are equal in length.

• Regular polygons are symmetrical.

Some examples of regular polygons are equilateral triangles, squares, regular pentagons and regular hexagons.

Irregular polygons

A polygon with sides and angles that are not equal in length is called an Irregular polygon.

Key properties of irregular polygons are:

• All interior angles are not equal in measurement.

• All sides are not equal in length.

• Irregular polygons are not symmetrical.

Some examples of irregular polygons are scalene triangles, irregular pentagons and irregular hexagons.

Convex polygons

A polygon with interior angles less than 180° is called a Convex polygon. In these polygons, no vertices point inwards.
Key properties of Convex polygons are:

• All diagonals lie inside the polygon.

• No line segment between two vertices goes outside the polygon.

Some examples of Convex polygons are equilateral triangles, squares, and other polygons where interior angles are less than 180°.

Concave polygons

A polygon with at least one interior angle greater than 180° is called a Concave polygon. In these polygons, at least one vertex points inward.
Key properties of Concave polygons are:

• At least one diagonal lies outside the polygon.

• A line segment between two vertices can go outside the polygon.

Some examples of Concave polygons are star-shaped polygons and some irregularly shaped polygons.

Properties of Polygons

Every polygon has certain geometric properties that help us understand its shape, structure, and behavior. These properties are based on the sides, vertices, angles, diagonals, and symmetry of the polygon. Understanding these polygon properties is essential for solving geometry, mensuration, and quantitative aptitude questions.

Sides of a Polygon

The straight line segments that form the boundary of a polygon are called its sides.

A polygon must have at least three sides.

Examples

• Triangle → 3 sides
• Quadrilateral → 4 sides
• Pentagon → 5 sides
• Hexagon → 6 sides
• Octagon → 8 sides

Key Facts

• All sides are straight line segments.
• Sides meet only at their endpoints.
• The number of sides determines the type of polygon.

Vertices of a Polygon

The points where two sides of a polygon meet are called vertices.

The singular form of vertices is vertex.

Examples

• Triangle → 3 vertices
• Pentagon → 5 vertices
• Hexagon → 6 vertices

Key Facts

• Every polygon has the same number of vertices as sides.
• Interior and exterior angles are formed at the vertices.

Angles of a Polygon

The angle formed between two adjacent sides of a polygon is called an interior angle.

Examples

A regular pentagon has:

• 5 sides
• 5 interior angles

A regular hexagon has:

• 6 sides
• 6 interior angles

Key Facts

• Every polygon has interior angles.
• Closed polygons also have exterior angles.
• The sum of angles depends on the number of sides.

Diagonals of a Polygon

A diagonal is a line segment joining two non-adjacent vertices of a polygon.

Examples

• A quadrilateral has 2 diagonals.
• A pentagon has 5 diagonals.
• A hexagon has 9 diagonals.

Key Facts

• Diagonals lie inside the polygon.
• They divide polygons into triangles.
• They help in area calculations and geometric proofs.

Symmetry in Polygons

Symmetry refers to the balanced arrangement of a shape such that one half mirrors the other.

Types of Symmetry

• Line Symmetry
• Rotational Symmetry

Examples

Key Facts

• Regular polygons have maximum symmetry.
• Irregular polygons may have little or no symmetry.

Polygon Formulas

Polygon formulas are used to calculate angles, diagonals, perimeter, and area. These formulas are among the most important geometry formulas for school mathematics and competitive exams.

Sum of Interior Angles Formula

The sum of all interior angles of an $n$-sided polygon is:

$\text{Sum of Interior Angles}=(n-2)\times180^\circ$

where:

$n=$ Number of sides

Example

For a pentagon:

$(5-2)\times180^\circ$

$=540^\circ$

Therefore, the sum of interior angles of a pentagon is $540^\circ$.

Exterior Angle Formula

The sum of all exterior angles of any polygon is always:

$\text{Sum of Exterior Angles}=360^\circ$

Example

For a regular hexagon:

$\text{Each Exterior Angle}=\frac{360^\circ}{6}$

$=60^\circ$

Number of Diagonals Formula

The number of diagonals in an $n$-sided polygon is:

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

Example

For a hexagon:

$=\frac{6(6-3)}{2}$

$=\frac{18}{2}$

$=9$

Therefore, a hexagon has 9 diagonals.

Perimeter Formula

The perimeter of a polygon is the total length of all its sides.

For a regular polygon:

$\text{Perimeter}=n\times s$

where:

$n=$ Number of sides

$s=$ Length of each side

Example

For a regular pentagon with side length 8 cm:

$\text{Perimeter}=5\times8$

$=40\text{ cm}$

Area Formula of Regular Polygon

The area of a regular polygon is:

$\text{Area}=\frac{1}{2}\times\text{Perimeter}\times\text{Apothem}$

or

$\text{Area}=\frac{1}{2}Pa$

where:

$P=$ Perimeter

$a=$ Apothem

This formula is commonly used for regular polygons such as pentagons, hexagons, and octagons.

Area of Polygons

The area of a polygon represents the amount of space enclosed within its boundary. Different types of polygons have different area formulas.

Area of Triangle

The area of a triangle is:

$\text{Area}=\frac{1}{2}\times\text{Base}\times\text{Height}$

Example

Base = 8 cm

Height = 5 cm

$\text{Area}=\frac{1}{2}\times8\times5$

$=20\text{ cm}^2$

Area of Quadrilateral

The area formula depends on the type of quadrilateral.

Rectangle

$\text{Area}=\text{Length}\times\text{Breadth}$

Square

$\text{Area}=\text{Side}^2$

Example

For a square of side 6 cm:

$\text{Area}=6^2$

$=36\text{ cm}^2$

Area of Regular Polygon

For any regular polygon:

$\text{Area}=\frac{1}{2}\times\text{Perimeter}\times\text{Apothem}$

Example

Perimeter = 30 cm

Apothem = 8 cm

$\text{Area}=\frac{1}{2}\times30\times8$

$=120\text{ cm}^2$

Area Using Apothem and Perimeter

The apothem-perimeter formula is particularly useful for regular polygons.

$\text{Area}=\frac{1}{2}Pa$

where:

$P=$ Perimeter

$a=$ Apothem

This method avoids dividing the polygon into multiple triangles.

Interior and Exterior Angles of Polygons

Interior and exterior angles are among the most important concepts in polygon geometry. They help determine the shape and properties of a polygon.

Interior Angles Explained

An interior angle is the angle formed inside a polygon by two adjacent sides.

Formula for Each Interior Angle of a Regular Polygon

$\text{Interior Angle}=\frac{(n-2)\times180^\circ}{n}$

Example

For a regular hexagon:

$\frac{(6-2)\times180^\circ}{6}$

$=\frac{720^\circ}{6}$

$=120^\circ$

Exterior Angles Explained

An exterior angle is formed when one side of a polygon is extended.

Formula for Each Exterior Angle of a Regular Polygon

$\text{Exterior Angle}=\frac{360^\circ}{n}$

Example

For a regular octagon:

$\frac{360^\circ}{8}$

$=45^\circ$

Relationship Between Interior and Exterior Angles

At every vertex of a polygon:

$\text{Interior Angle}+\text{Exterior Angle}=180^\circ$

Example

If the interior angle is:

$140^\circ$

Then,

$\text{Exterior Angle}=180^\circ-140^\circ$

$=40^\circ$

Solved Examples

Example 1

Find the sum of interior angles of a decagon.

Using:

$\text{Sum of Interior Angles}=(n-2)\times180^\circ$

$=(10-2)\times180^\circ$

$=1440^\circ$

Example 2

Find each exterior angle of a regular pentagon.

$\text{Exterior Angle}=\frac{360^\circ}{5}$

$=72^\circ$

How to Identify a Polygon?

Identifying polygons correctly is important because many geometric figures may appear similar but are not polygons.

Characteristics of a Polygon

A figure is a polygon if:

• It is closed.
• It is two-dimensional.
• It has three or more sides.
• All sides are straight line segments.
• Adjacent sides intersect only at endpoints.

Examples of Polygons

• Triangle
• Square
• Rectangle
• Pentagon
• Hexagon

Open vs Closed Figures

A polygon must always be a closed figure.

Closed Figure

A triangle is closed because all sides connect completely.

Open Figure

A shape with a gap between two endpoints is not a polygon.

Polygon vs Non-Polygon Shapes

Not every geometric shape is a polygon.

Quick Identification Rule

A shape is a polygon if it is:

• Closed
• Flat (2D)
• Made entirely of straight line segments

If any side is curved or the figure is open, it is not a polygon.

Difference Between Regular and Irregular Polygons

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.

Best Books for Polygons and Geometry

Polygons form a major part of geometry and mensuration. The following books are useful for understanding polygon properties, angles, area formulas, and exam-oriented questions.

Shortcut Tips and Tricks for Polygon Questions

Given below are the shortcut tips and tricks which are related to polygon questions:

Important Polygon Formula Table

Given below are the important formulae which are related to polygon questions:

Common Polygon Names Table

These tables provide quick revision material for divisibility rules, vector scalar multiplication, and polygon geometry formulas, making them useful for school exams, board exams, JEE, CUET, SSC, Banking, Railways, and other competitive examinations.

Practice Questions based on Polygons

Q1. If the exterior angle of a regular polygon is $18^\circ$, then the number of diagonals in the polygon is:

1. 180
2. 170
3. 150
4. 140

Hint: Use:

$\text{Number of sides}=\frac{360^\circ}{\text{Exterior Angle}}$

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

Solution:

Given,

Each exterior angle of the regular polygon is $18^\circ$.

We know,

$\text{Number of sides}=\frac{360^\circ}{18^\circ}$

$=\frac{360}{18}$

$=20$

Thus, the polygon has 20 sides.

Now,

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

$=\frac{20(20-3)}{2}$

$=\frac{20\times17}{2}$

$=10\times17$

$=170$

Correct Answer: 170

Q2. There are two regular polygons with numbers of sides equal to $(n-1)$ and $(n+2)$. Their exterior angles differ by $6^\circ$. The value of $n$ is:

1. 14
2. 12
3. 11
4. 13

Hint: Use:

$\text{Exterior Angle}=\frac{360^\circ}{\text{Number of Sides}}$

Solution:

Given,

The numbers of sides are $(n-1)$ and $(n+2)$.

Their exterior angles differ by $6^\circ$.

Therefore,

$\frac{360}{n-1}-\frac{360}{n+2}=6$

Taking 360 common,

$360\left(\frac{1}{n-1}-\frac{1}{n+2}\right)=6$

$360\left(\frac{n+2-(n-1)}{(n-1)(n+2)}\right)=6$

$360\left(\frac{3}{(n-1)(n+2)}\right)=6$

$\frac{1080}{(n-1)(n+2)}=6$

$(n-1)(n+2)=180$

$n^2+n-2=180$

$n^2+n-182=0$

$(n+14)(n-13)=0$

$n=-14$ or $n=13$

Since the number of sides cannot be negative,

$n=13$

Correct Answer: 13

Q3. How many diagonals are there in an octagon?

1. 12
2. 14
3. 20
4. 24

Hint: Use:

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

Solution:

Given,

Number of sides of an octagon,

$n=8$

Using the formula,

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

$=\frac{8(8-3)}{2}$

$=\frac{8\times5}{2}$

$=4\times5$

$=20$

Correct Answer: 20

Q4. If one of the interior angles of a regular polygon is $\frac{15}{16}$ times one of the interior angles of a regular decagon, then find the number of diagonals of the polygon.

1. 20
2. 14
3. 2
4. 35

Hint: Use:

$\text{Interior Angle}=\frac{(n-2)\times180^\circ}{n}$

$\text{Number of Diagonals}=\frac{n(n-3)}{2}$

Solution:

Interior angle of a regular decagon:

$=\frac{(10-2)\times180^\circ}{10}$

$=\frac{8\times180^\circ}{10}$

$=144^\circ$

Given,

Interior angle of the required polygon

$=\frac{15}{16}\times144^\circ$

$=135^\circ$

Now,

$\frac{(n-2)\times180^\circ}{n}=135^\circ$

$180n-360=135n$

$45n=360$

$n=8$

Thus, the polygon has 8 sides.

Now,

$\text{Number of Diagonals}=\frac{8(8-3)}{2}$

$=\frac{8\times5}{2}$

$=20$

Correct Answer: 20

Q5. The length and breadth of a rectangle are 8 cm and 6 cm respectively. The rectangle is cut on its four vertices such that the resulting figure is a regular octagon. What is the length of the side (in cm) of the octagon?

1. $3\sqrt{11}-7$
2. $5\sqrt{13}-8$
3. $5\sqrt{7}-11$
4. $6\sqrt{11}-9$

Hint: Let the side of the octagon be $a$ cm and apply the Pythagoras theorem.

Solution:

Given,

Length of rectangle $=8$ cm

Breadth of rectangle $=6$ cm

Let the side of the regular octagon be $a$ cm.

Using the geometry of the figure,

$a^2=\frac{(8-a)^2}{4}+\frac{(6-a)^2}{4}$

Multiplying both sides by 4,

$4a^2=(8-a)^2+(6-a)^2$

$4a^2=(64-16a+a^2)+(36-12a+a^2)$

$4a^2=100-28a+2a^2$

$2a^2+28a-100=0$

Dividing by 2,

$a^2+14a-50=0$

Using the quadratic formula,

$a=\frac{-14+\sqrt{14^2-4(1)(-50)}}{2}$

$=\frac{-14+\sqrt{196+200}}{2}$

$=\frac{-14+\sqrt{396}}{2}$

$=\frac{-14+6\sqrt{11}}{2}$

$=3\sqrt{11}-7$

Therefore,

$\text{Side of the octagon}=3\sqrt{11}-7$ cm

Correct Answer: $3\sqrt{11}-7$

Q6. ABCDEF is a regular hexagon. The side of the hexagon is 36 cm. What is the area of $\triangle ABC$?

1. $324\sqrt{3}\ \text{cm}^2$
2. $360\sqrt{3}\ \text{cm}^2$
3. $240\sqrt{3}\ \text{cm}^2$
4. $192\sqrt{3}\ \text{cm}^2$

Hint: A regular hexagon can be divided into 6 congruent equilateral triangles.

Solution:

Given,

ABCDEF is a regular hexagon.

Side of the hexagon,

$a=36$ cm

Area of a regular hexagon:

$=6\times\frac{\sqrt{3}}{4}a^2$

Substituting $a=36$,

$=6\times\frac{\sqrt{3}}{4}\times(36)^2$

Since the hexagon consists of 6 equal equilateral triangles,

$\text{Area of }\triangle ABC$

$=\frac{1}{6}\times6\times\frac{\sqrt{3}}{4}\times(36)^2$

$=\frac{\sqrt{3}}{4}\times1296$

$=324\sqrt{3}\ \text{cm}^2$

Correct Answer: $324\sqrt{3}\ \text{cm}^2$

Q7. PQRS is a square whose side is 16 cm. What is the value of the side (in cm) of the largest regular octagon that can be cut from the given square?

1. $8-4\sqrt{2}$
2. $16+8\sqrt{2}$
3. $16\sqrt{2}-16$
4. $16-8\sqrt{2}$

Hint: Apply the Pythagoras theorem in the corner triangles.

Solution:

Given,

PQRS is a square of side 16 cm.

Let,

$ER=RD=a$

Then,

$ED=CD=(16-2a)$

From right-angled triangle ERD,

$(16-2a)^2=a^2+a^2$

$(16-2a)^2=2a^2$

Expanding,

$256-64a+4a^2=2a^2$

$2a^2-64a+256=0$

Dividing by 2,

$a^2-32a+128=0$

Using the quadratic formula,

$a=\frac{32\pm\sqrt{32^2-4(1)(128)}}{2}$

$=\frac{32\pm\sqrt{1024-512}}{2}$

$=\frac{32\pm\sqrt{512}}{2}$

$=\frac{32\pm16\sqrt{2}}{2}$

$=16\pm8\sqrt{2}$

Since $a<16$,

$a=16-8\sqrt{2}$

Now,

$\text{Side of octagon}$

$=16-2a$

$=16-2(16-8\sqrt{2})$

$=16-32+16\sqrt{2}$

$=16\sqrt{2}-16$

Correct Answer: $16\sqrt{2}-16$

Q8. If the sum of the measures of all the interior angles of a polygon is $1440^\circ$, find the number of sides of the polygon.

1. 8
2. 12
3. 10
4. 14

Hint: Use:

$\text{Sum of Interior Angles}=(n-2)\times180^\circ$

Solution:

Given,

Sum of interior angles

$=1440^\circ$

We know,

$(n-2)\times180^\circ=1440^\circ$

Dividing both sides by $180^\circ$,

$n-2=8$

Adding 2 to both sides,

$n=10$

Therefore, the polygon has 10 sides.

Correct Answer: 10

Q9. The sum of all exterior angles of a polygon is:

1. $180^\circ$
2. $270^\circ$
3. $360^\circ$
4. Depends on the number of sides

Hint: This is a standard property of all polygons.

Solution:

For any polygon, whether regular or irregular, the sum of one exterior angle at each vertex is always:

$360^\circ$

This property is independent of the number of sides.

Therefore,

$\text{Sum of Exterior Angles}=360^\circ$

Correct Answer: $360^\circ$

Q10. Find the sum of the interior angles of a 12-sided polygon.

1. $1620^\circ$
2. $1800^\circ$
3. $1440^\circ$
4. $1260^\circ$

Hint: Use:

$\text{Sum of Interior Angles}=(n-2)\times180^\circ$

Solution:

Given,

$n=12$

Using the formula,

$\text{Sum of Interior Angles}=(12-2)\times180^\circ$

$=10\times180^\circ$

$=1800^\circ$

Correct Answer: $1800^\circ$

Related Quantitative Aptitude Topics

To gain a deeper understanding of quantitative aptitude, it is helpful to learn other important mathematics topics that complement this concept. These topics are widely used in aptitude tests and play a key role in strengthening fundamental mathematical skills.