Polygons - Definition, Types, Shapes and Properties

Have you ever looked at a stop sign, a honeycomb, or even a slice of pizza and wondered what shape it is? All of these are examples of polygons. A polygon is a 2D shape made up of straight lines that close together to form a figure.

On this page, we will explore everything about polygons - what they are, the different types, their properties with examples.

By the end of this page, you will be able to:

• Define a polygon and spot polygon shapes around you.
• Tell the difference between a regular and an irregular polygon.
• Name and describe different polygon shapes and their properties.
• Calculate the number of diagonals in any polygon using a simple formula.

Each concept is explained using examples and diagrams.

Table of Contents

• What Is a Polygon?
• Types of Polygons
• Polygonal Shape Names
• Triangle Polygon
• Types of Triangles
• Quadrilateral Polygon
• Square Polygon
• Rectangle Polygon
• Rhombus Polygon
• Parallelogram Polygon
• Trapezium Polygon
• Pentagon Polygon
• Hexagon Polygon
• Octagon Polygon
• Number of Diagonals in a Polygon
• Solved examples on Polygons
• Practice questions on Polygons
• Things You Have Learnt

What Is a Polygon?

A polygon is a two-dimensional closed figure in geometry, made by joining three or more straight lines end to end.

Example - Drawing a shape without lifting your pencil - and connecting the end where it started. That closed figure, as long as it has straight sides, is a polygon.

• A polygon must be a closed figure - all the sides must connect.
• A polygon must have only straight sides
• A polygon must have at least three sides.
  
  

Every polygon is a closed figure, but not every closed figure is a polygon.

Look at the shapes below:

These are all closed figures, but some of their sides are curved. Since polygons can only have straight sides, these shapes are not polygons.

Types of Polygons

Depending on the length of their sides and the size of their angles, polygons are divided into two types (types of polygons):

• Regular Polygon
• Irregular Polygon
  
  

What Is a Regular Polygon?

A regular polygon is a shape where every side is exactly the same length and every angle is exactly the same size - it is symmetrical.

A square is a great example of a regular polygon - all four sides are equal and all four angles are 90°.

What Is an Irregular Polygon?

An irregular polygon is a shape where the sides are all different lengths and the angles are all different sizes.

Most shapes we see in real life - like the outline of a country on a map - are irregular polygons.

Read more:

• Class 9 - Frequency Polygon
• Class 8 - Types of Polygon
• Class 5 - Polygons
  
  

Polygonal Shape Names

Polygons are named based on the number of sides they have.

 

A quick tip to remember - the names come from Greek and Latin numbers. Tri means three, quad means four, penta means five, hex means six, and so on!

Triangle Polygon

The triangle is the simplest polygon of all. The following are its properties:

• A triangle is a three-sided closed figure.
• The three line segments that form a triangle are called its sides.
• The points where two sides meet are called vertices (singular: vertex).
• The angle formed at each vertex is called an angle of the triangle.
• No matter what a triangle looks like, the sum of its three angles is always 180°.
• Every triangle has 3 sides, 3 vertices, and 3 angles.
  
  

In the figure above, ABC is a triangle:

• Sides = AB, AC, and BC
• Vertices = A, B, and C
• Angles = ∠A, ∠B, and ∠C
• ∠A + ∠B + ∠C = 180°
  
  

Types of Triangles

Triangles come in three different types depending on the length of their sides:

• Equilateral Triangle
• Isosceles Triangle
• Scalene Triangle

Equilateral Triangle

The word equilateral means "equal sides." In an equilateral triangle, all three sides are the same length and all three angles are the same size.

• Each angle of an equilateral triangle is always 60° - because 60° × 3 = 180°.
• An equilateral triangle is also called a regular triangle.
• It is perfectly symmetrical.

 

In triangle ABC:

• Sides: AB = BC = AC
• Vertices: A, B, and C
• Angles: ∠A = ∠B = ∠C = 60°

Isosceles Triangle

The word isosceles comes from Greek meaning "equal legs." In an isosceles triangle, exactly two sides are equal and the two base angles are equal.

In triangle AOB:

• Sides: AO = AB, with OB being different
• Vertices: A, O, and B
• Angles: ∠O = ∠B and ∠A + ∠O + ∠B = 180°
  
  

Scalene Triangle

A scalene triangle is the most "irregular" of the three - all three sides are different lengths and all three angles are different.

No sides are equal, no angles are equal - but the angles still always add up to 180°.

 

In triangle ABC:

• Sides: AB ≠ AC ≠ BC
• Vertices: A, B, and C
• Angles: ∠A ≠ ∠B ≠ ∠C and ∠A + ∠B + ∠C = 180°
  
  

Read more on interior angles of a polygon

Quadrilateral Polygon

Any polygon with four sides is called a quadrilateral. The word comes from Latin - quadri means four and latus means side.

• Every quadrilateral has 4 sides, 4 vertices, and 4 angles.
• The sum of all interior angles of any quadrilateral is always 360°.
• There are several types of quadrilaterals - square, rectangle, rhombus, parallelogram, and trapezium.
  
  

Square Polygon

A square is one of the most familiar shapes in the world - from floor tiles to chessboards.

• A square is a regular quadrilateral - all four sides are equal and all four angles are equal.
• Opposite sides of a square are parallel to each other.
• Each interior angle is exactly 90° (a right angle).
• The sum of all interior angles is 360°.
  
  

In the figure above, ABCD is a square:

• Sides: AB = BC = CD = DA
• Vertices: A, B, C, and D
• Angles: ∠A = ∠B = ∠C = ∠D = 90°
• Sum of angles = 360°
  
  

Rectangle Polygon

A rectangle is like a stretched-out square. The opposite sides are equal but not all four sides are the same length.

• A rectangle has 4 sides where opposite sides are parallel and equal.
• All four interior angles are 90°, just like a square.
• The sum of all interior angles is 360°.
• Every square is a rectangle, but not every rectangle is a square.
  
  

In the figure above, ABCD is a rectangle:

• Sides: AB = CD and AD = BC
• Vertices: A, B, C, and D
• Angles: ∠A = ∠B = ∠C = ∠D = 90°
• Sum of angles = 360°

Rhombus Polygon

A rhombus looks like a tilted square. It is sometimes called a diamond shape.

• A rhombus has 4 equal sides - just like a square.
• But unlike a square, the angles of a rhombus are not all 90°.
• Opposite sides are parallel and opposite angles are equal.
• The diagonals of a rhombus cross each other at 90° and bisect each other.
• The sum of all interior angles is 360°.
  
  

Remember: A square is actually a special rhombus where all the angles happen to be 90°

Parallelogram Polygon

A parallelogram is a quadrilateral where opposite sides run parallel to each other - like two pairs of train tracks.

• Opposite sides are parallel and equal in length.
• Opposite angles are equal.
• Adjacent angles (angles next to each other) always add up to 180°.
• The diagonals bisect each other - they cut each other exactly in half.
• The sum of all interior angles is 360°.
  
  

Remember: A rectangle and a square are both special types of parallelograms

Trapezium Polygon

A trapezium is a quadrilateral that has exactly one pair of parallel sides.

• The two parallel sides are called the bases.
• The two non-parallel sides are called the legs.
• The sum of all interior angles is 360°.
• If the two legs are equal in length, it is called an isosceles trapezium.
  
  

Pentagon Polygon

Move up to five sides and you have a pentagon. The name comes from the Greek word penta meaning five.

• A pentagon has 5 sides, 5 vertices, and 5 angles.
• The sum of all interior angles of a pentagon is 540°.
• In a regular pentagon, all five sides are equal and each angle is 108°.
• A regular pentagon has 5 diagonals.
  
  

Hexagon Polygon

A hexagon has six sides.

• A hexagon has 6 sides, 6 vertices, and 6 angles.
• The sum of all interior angles of a hexagon is 720°.
• In a regular hexagon, all six sides are equal and each angle is 120°.
• A regular hexagon has 9 diagonals.
  
  

Octagon Polygon

An octagon has eight sides.

• An octagon has 8 sides, 8 vertices, and 8 angles.
• The sum of all interior angles of an octagon is 1080°.
• In a regular octagon, all eight sides are equal and each angle is 135°.
• A regular octagon has 20 diagonals.
  
  

Number of Diagonals in a Polygon

A diagonal is a straight line that connects two vertices of a polygon that are not next to each other.

• A triangle has no diagonals at all - because every vertex is directly next to the other two.
• As the number of sides increases, the number of diagonals also increases
  
  

 

Number of diagonals formula:

Number of diagonals = n(n - 3) ÷ 2, where n is the number of sides.

Let's try it: For a hexagon, n = 6
So, 6 × (6−3) ÷ 2 = 6 × 3 ÷ 2 = 9 diagonals

For an octagon, n = 8
So, 8 × (8−3) ÷ 2 = 8 × 5 ÷ 2 = 20 diagonals

Solved Examples on Polygons

Example 1: Which of the following is a polygon - a circle, a triangle, or a shape with one curved side?

Solution:

• A circle has a curved boundary and no straight sides. It is not a polygon.
• A shape with one curved side does not meet the condition of having all straight sides. It is not a polygon.
• A triangle is a closed figure with three straight sides. It is a polygon.

Answer: A triangle is a polygon.

Example 2: Find the sum of the interior angles of a hexagon.

Solution: The formula for the sum of interior angles is (n - 2) × 180°, where n is the number of sides.

For a hexagon, n = 6.

Sum of interior angles = (6 - 2) × 180° = 4 × 180° = 720°

Answer: The sum of interior angles of a hexagon is 720°.

Example 3: Finding the Number of Diagonals

Question: How many diagonals does a pentagon have?

Solution: The formula for the number of diagonals is n(n - 3) ÷ 2, where n is the number of sides.

For a pentagon, n = 5.

Number of diagonals = 5 × (5−3) ÷ 2 = 5 × 2 ÷ 2 = 5 diagonals

Answer: A pentagon has 5 diagonals.

Example 4: A quadrilateral has sides of lengths 5 cm, 5 cm, 5 cm, and 5 cm, and all angles are equal to 90°. Is it a regular or irregular polygon?

Solution:

• All four sides are equal in length (5 cm each).
• All four angles are equal (90° each).
• Since all sides and all angles are equal, this quadrilateral is a regular polygon.
• A quadrilateral with four equal sides and four equal angles of 90° is a square.

Answer: It is a regular polygon - specifically a square.

Example 5: Find the measure of each interior angle of a regular octagon.

Solution: Step 1 - Find the sum of interior angles using (n−2) × 180°, where n = 8.

Sum of interior angles = (8−2) × 180° = 6 × 180° = 1080°

Step 2 - Divide by the number of angles to find each angle.

Each interior angle = 1080° ÷ 8 = 135°

Answer: Each interior angle of a regular octagon is 135°.

Practice Questions on Polygons

Q1. Name the polygon that has the least number of sides. How many sides does it have?

Q2. What is the sum of interior angles of a pentagon? Use the formula to calculate.

Q3. A polygon has 7 sides. What is it called? Calculate the sum of its interior angles.

Q4. How many diagonals does a quadrilateral have? Verify using the diagonal formula.

Q5. A regular polygon has each interior angle equal to 120°. Identify the polygon.

Q6. A floor tile is in the shape of a regular hexagon. How many diagonals can be drawn inside the tile?

Q7. Look at the shapes below and identify which ones are polygons and which are not. Give a reason for each.

• A rectangle
• A semicircle
• An oval
• A rhombus
• A shape with 6 straight sides that is fully closed

Q8. A polygon has 10 sides. Find:

• The sum of its interior angles
• The number of diagonals
• The measure of each interior angle if it is a regular polygon

Q9. Two polygons are described below. Identify each one:

• Polygon A has 4 equal sides, opposite sides are parallel, and all angles are 90°.
• Polygon B has 4 equal sides, opposite sides are parallel, but the angles are not 90°.

Q10. A stop sign has 8 sides of equal length and 8 equal angles. A yield sign is triangular.

• What type of polygon is each sign?
• Are they regular or irregular polygons?
• Calculate the sum of interior angles for each sign.

Things You Have Learnt

• A polygon is a 2D closed figure with three or more straight sides.
• Polygons with equal sides and angles are regular. Those with unequal sides and angles are irregular.
• The simplest polygon is a triangle with just 3 sides.
• The sum of interior angles formula is (n−2) × 180°.
• The number of diagonals formula is n(n−3) ÷ 2.
• Polygons are everywhere in real life - honeycomb (hexagon), tiles (square), and even the buildings around you.