Chapter Contents

Curve, Polygon and Types of Polygon

Introduction

Polygon term originates from the two Greek words “Poly” and “Gon”, where “Poly” stands for “many” and “Gon” stands for angle. So the complete word “Polygon” deals with shapes having many angles.

Interestingly, the number of angles in a shape is always equal to the number of edges or sides it may have. In short, we can say a polygon is a shape with many sides or angles.

So, what are the various types of polygons in geometry and how are they made? Not to go so far, we use curves to make various shapes in geometry. So, to draw all polygons we use curves. Polygon has its angles, edges, vertices and diagonals. We can visualise them and see how many are they in any polygon.

What is a Curve?

Curve is a plane figure which is made by joining a number of points without lifting a pencil from paper and without retracing any portion of drawing.
Moreover, a curve which does not intersect itself is called open curve and on the other hand, a curve which intersect itself is called closed curve.

Curve in geometry

What is a Polygon in Geometry?

Polygon is a closed curve which is made up of only finite number of line segments and any two line segments do not intersect each other except at their end points. It is a two dimensional figure in geometry.
The line segments of a polygon are called sides or edges and end points of the line segments are called its vertices or corners.

Different types of polygon

What are Adjacent Sides?

Any two sides of a polygon with a common end point are called adjacent sides of the polygon.

What are Adjacent Vertices?

The end points of a side of a polygon are called adjacent vertices.

What is a Diagonal of polygon?

A diagonal is a line segment which is formed by joining two vertices which are not adjacent.

Example

AC is a diagonal.
BD is a diagonal.

Diagonals

Types of polygon

Convex polygon

A polygon is said to be a convex polygon, if its diagonals completely lie inside of it. Its all interior angles are less than 180 degree.

Convex Polygon

Concave polygon

A polygon is said to be a concave polygon, if its diagonals completely lie outside of it. Its one or more than one interior angle is more than 180 degrees.

Concave Polygon

Regular polygon

A regular polygon is a polygon whose all sides and all angles are equal. In other words we can say, it is both equilateral and equiangular.

Example

Square is a regular polygon as its sides are equal and all angles are equal.

Regular Polygon

Formula

Interior angle of regular polygon
= (n-2) × 180°
or
= (n-2) × π radian

Irregular polygon

Polygon which is not a regular polygon, i.e. if it is not equiangular and not equilateral is called irregular polygon.

Example

Rectangle is an irregular polygon.

Irregular Polygon

Simple polygon

It is a flat shape consisting of straight and non intersecting line segments that are joined to form a single close path. A Simple Polygon does not cross over itself. It is also called Jordan Polygon.
All convex polygons are simple.

Simple Polygon

Classification of polygon based on number of sides

Number of sides of polygon	Name of polygon	Number of angles of polygon
3	Triangle	3
4	Quadrilateral	4
5	Pentagon	5
6	Hexagon	6
7	Heptagon	7
8	Octagon	8
9	Nonagon	9
10	Decagon	10

Frequently Asked Questions

1) What is a polygon?

A closed plane figure formed by line segments and two line segments which intersect at only their end points.

2) What is sum of angles of polygon?

The sum of angles of polygon with n sides is (n-2)180⁰.

3) What is the sum of exterior angles of polygon?

Sum of exterior angles of polygon is 360⁰.

4) Is equilateral triangle a regular polygon?

Yes, equilateral triangle is a regular polygon as its all sides are equal and all its angles are equal.

5) Is rectangle a regular polygon?

No, rectangle is not a regular polygon as its angles are equal but all sides are not equal.

Solved Examples

1) Find the number of sides of polygon if each interior angle is 140⁰.

Solution

Let number of sides of polygon = n

Each interior angle = $\frac{(n-2)180}{n}$

140 = $\frac{(n-2)180}{n}$

140n = 180n - 360

360 = 180n - 140n

360 = 40n

$\frac{360}{40}$ = n

n = 9

2) Find the number of sides of regular polygon if each exterior angle is 40⁰

Solution

Sum of all exterior angle = 360⁰

Each exterior angle = 40⁰

Number of exterior angles = $\frac{360}{40}$

Number of exterior angles = 9

∴ Number of sides of polygon = 9

3) If each interior angle of regular polygon is 150⁰ then find its exterior angle.

Solution

Each interior angle = 150⁰.

We know that, interior angles + exterior angles = 180⁰

150⁰ + exterior angles = 180⁰

exterior angles = 180⁰ - 150⁰

exterior angles = 30⁰

4) The ratio between exterior angle and interior angle of a regular polygon is 1:5. Find the number of sides of polygon.

Solution

Let exterior angle and interior angle be x and 5x respectively.

Also, interior angles + exterior angles = 180⁰

5x + x = 180⁰

6x = 180⁰

x = $\frac{180}{6}$

x = 30⁰

∴ exterior angle = 30⁰

interior angle = 5x = 5 × 30⁰

i.e. interior angle = 150⁰

Worksheet 1

Download

Multiple choice questions

1) A polygon with 5 sides is called as

a) hexagon

b) decagon

c) pentagon

d) nonagon

2) A polygon with 8 sides is called as

a) hexagon

b) heptagon

c) octagon

d) decagon

3) In convex polygon, each angle of polygon is

a) = 90⁰

b) = 180⁰

c) < 90⁰

d) < 180⁰

4) In concave polygon, at least one angle of polygon is

a) < 90⁰

b) < 180⁰

c) > 90⁰

d) > 180⁰

5) A polygon is said to be regular polygon, if all of its

a) interior angles are equal

b) exterior angles are equal

c) sides are equal

d) all of these

6) At each vertex of polygon, the sum of interior angle and exterior angle is equal to

a) 90⁰

b) 180⁰

c) 270⁰

d) 360⁰

7) The regular polygon with three sides is a

a) isosceles triangle

b) equilateral triangle

c) scalene triangle

d) rhombus

8) The least number of sides in a polygon is

a) 2

b) 4

c) 1

d) 3

9) The sum of interior angles of a polygon with 7 sides is

a) 900°

b) 800°

c) 700°

d) 540°

10) If the sum of interior angles of a polygon is 1440° then the number of sides of the polygon are

a) 15

b) 10

c) 11

d) 14

MCQ Answer Key Hide Show

Worksheet 2

Fill in the blanks

1) In a regular polygon all sides and all ___________ are equal.

2) ___________ triangle is a regular polygon.

3) A triangle has no ___________ .

4) Sum of exterior angles of a polygon is ___________ .

5) Concave polygon has at least one angle more than ___________ .

6) Sum of interior angles of a pentagon is ___________ .

7) The number of sides of a polygon must be greater than or equal to ___________ .

8) The sum of interior angles of eight sided polygon is ___________ .

9) A regular hexagon has ___________ diagonals.

10) A polygon whose sum of interior angles is 1440° has ___________ sides.

Help box

540°

180°

1080°

equilateral

diagonal

360°

angles

Worksheet 3

Write True or False in the boxes.

1)	A square is a regular polygon.	▭
2)	A triangle has three diagonals.	▭
3)	Sum of exterior angles of a regular hexagon is 720°	▭
4)	A n-sided convex polygon has $\frac{n (n - 3)}{2}$ diagonals.	▭
5)	Right angled triangle is a regular polygon.	▭
6)	In convex polygon each angle is less than 180°.	▭
7)	A heptagon has eight sides.	▭
8)	Each interior angle of a nonagon is 140°.	▭
9)	The number of sides in a polygon must be a whole number.	▭
10)	A decagon has 35 diagonals.	▭

Worksheet 4

Match the following.

S.N.	Name of polygon	Number of sides of polygon
1)	Pentagon	7
2)	Octagon	9
3)	Triangle	10
4)	Heptagon	6
5)	Nonagon	5
6)	Decagon	8
7)	Hexagon	3

Worksheet 5

Fill in the blanks.

S.N.	Number of sides of polygon	Sum of interior angles of polygon
1)	12	___________
2)	___________	1620°
3)	10	___________
4)	___________	1800°
5)	5	___________

Last updated on: 30-06-2024

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Triangle and the Types of Triangle

Quadrilateral and Types of Quadrilateral

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Curve, Polygon and Types of Polygon

Introduction

What is a Curve?

What is a Polygon in Geometry?

What are Adjacent Sides?

What are Adjacent Vertices?

What is a Diagonal of polygon?

Types of polygon

Convex polygon

Concave polygon

Regular polygon

Irregular polygon

Simple polygon

Classification of polygon based on number of sides

Frequently Asked Questions

1) What is a polygon?

2) What is sum of angles of polygon?

3) What is the sum of exterior angles of polygon?

4) Is equilateral triangle a regular polygon?

5) Is rectangle a regular polygon?

Solved Examples

1) Find the number of sides of polygon if each interior angle is 1400.

2) Find the number of sides of regular polygon if each exterior angle is 400

3) If each interior angle of regular polygon is 1500 then find its exterior angle.

4) The ratio between exterior angle and interior angle of a regular polygon is 1:5. Find the number of sides of polygon.

Worksheet 1

Multiple choice questions

1) A polygon with 5 sides is called as

2) A polygon with 8 sides is called as

3) In convex polygon, each angle of polygon is

4) In concave polygon, at least one angle of polygon is

5) A polygon is said to be regular polygon, if all of its

6) At each vertex of polygon, the sum of interior angle and exterior angle is equal to

7) The regular polygon with three sides is a

8) The least number of sides in a polygon is

9) The sum of interior angles of a polygon with 7 sides is

10) If the sum of interior angles of a polygon is 1440° then the number of sides of the polygon are

Worksheet 2

Fill in the blanks

1) In a regular polygon all sides and all ___________ are equal.

2) ___________ triangle is a regular polygon.

3) A triangle has no ___________ .

4) Sum of exterior angles of a polygon is ___________ .

5) Concave polygon has at least one angle more than ___________ .

6) Sum of interior angles of a pentagon is ___________ .

7) The number of sides of a polygon must be greater than or equal to ___________ .

8) The sum of interior angles of eight sided polygon is ___________ .

9) A regular hexagon has ___________ diagonals.

10) A polygon whose sum of interior angles is 1440° has ___________ sides.

Worksheet 3

Write True or False in the boxes.

A square is a regular polygon.

A triangle has three diagonals.

Sum of exterior angles of a regular hexagon is 720°

A n-sided convex polygon has n (n - 3) 2 diagonals.

Right angled triangle is a regular polygon.

In convex polygon each angle is less than 180°.

A heptagon has eight sides.

Each interior angle of a nonagon is 140°.

The number of sides in a polygon must be a whole number.

A decagon has 35 diagonals.

Worksheet 4

Match the following.

Worksheet 5

Fill in the blanks.

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1) Find the number of sides of polygon if each interior angle is 140⁰.

2) Find the number of sides of regular polygon if each exterior angle is 40⁰

3) If each interior angle of regular polygon is 150⁰ then find its exterior angle.

A n-sided convex polygon has $\frac{n (n - 3)}{2}$ diagonals.