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# Classification

## Introduction

We often categorise various objects as well as numbers based on their certain characteristics. Similarly, numbers as well as objects are compared and contrasted with each other on the basis of their features or properties. Is it possible to give a name to this process of comparing and contrasting of objects and numbers? This is where the term “ classification “ steps in. Let us learn what we mean by classification some general classifications that are in place that we make use of in everyday life as well as mathematics.

## Definition

Classification is the term used to represent the comparison of objects and numbers based on their similarities and differences. In other words, Classification means sorting or arranging objects into groups on the basis of common features or characteristics that they possess.

## What is Classification?

We now know that the process of comparing and contrasting objects and numbers and categorising them accordingly is what we call classification. For example, geometrical shapes are classified based on different properties such as number of sides, angles etc. Similarly, the products obtained from plants that have seeds are classified as vegetables while the products from the remaining parts of the plants are classified as vegetables.  Let us now learn about some common ways of classification that we make use of in everyday life.

### Ways of Classification

There are different ways in which we may classify various objects. They may be classified on the basis of their shapes, size colour and even the number of objects contained in them. Therefore, some of the classifications are –

1. Classification on the basis of shape
2. Classification on the basis of size
3. Classification on the basis of number
4. Classification on the basis of colour

Let us discuss these classifications one by one.

## Classification on the Basis of Shape

We often use various geometric shapes such as triangles, squares, circles etc. in mathematics. Observe the following shapes. Can you see some similarities or differences in them?

These shapes can be classified into different categories, based on various characteristics.

The shapes can be

1. 2 – dimensional or
2. 3- dimensional

This categorisation is based on the dimensions of the shapes. Let us learn about them one by one.

## 2 – Dimensional Shapes

2 Dimensional shapes or 2D shapes are shapes that have only length and breadth. For example, the shapes such as rectangles and squares are 2-dimensional shapes. 2 – Dimensional shapes are further classified on the basis of the number of sides. Let us learn about them.

### Classification Based on the Number of Sides

Based on the number of sides of a figure, we can classify the shapes as –

#### Closed Shapes having no side

A common geometrical figure that is a closed shape and does have any sides is a circle. A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance from a fixed point in the plane. This fixed distance is called the radius of the circle and the fixed point is called the centre of the circle. Following is the example of a circle –

#### Closed Shapes Having 3 Sides – Triangle

A triangle is a polygon that is made up of 3 sides, 3 angles and three vertices.

We can notice that there are three sides of a triangle. Based on the length of these sides, triangles can further be classified as

1. Scalene Triangle
2. Isosceles Triangle
3. Equilateral Triangle

Scalene Triangle – A scalene triangle is a triangle in which all three sides are of unequal lengths. This means that in a scalene triangle, no two sides are equal. Following is an example of a scalene triangle –

Isosceles Triangle – A triangle is said to be an isosceles triangle if two of its sides are equal. Following is an example of an equilateral triangle –

Equilateral Triangle – A triangle is said to be an equilateral triangle if all its sides are equal. Following is an example of an equilateral triangle –

Triangles can further be classified based on the kind of angles in a triangle. These classifications are–

Right Angled Triangle A triangle is said to be a right-angled triangle if one of the angles of the triangle is a right angle, i.e. 90o. Suppose, we have a triangle, ABC where △ABC = 90o.  Then such a triangle is called a right-angled triangle which would be of a shape similar to the below figure.

Equilateral Triangle A triangle is said to be an equilateral triangle if all its sides are equal. Also, if all the three sides are equal in a triangle, the three angles are equal.

Isosceles Triangle A triangle is said to be an Isosceles triangle if its two sides are equal. If two sides are equal, then the angles opposite to these sides are also equal. For example, in the following triangle, AB = AC. Therefore ∆ABC is an Isosceles triangle.

∠B = ∠C

#### Closed Shapes Having 4 Sides

Closed shapes that are made of four sides are known as quadrilaterals. A quadrilateral is a closed shape that is formed by joining four points among which any three points are non-collinear. In other words, a quadrilateral is a polygon made up of four sides. The order of vertices needs to be kept in mind while naming a quadrilateral. Following is an example of a quadrilateral –

We have different types of quadrilateral according to the arrangement of these four sides. Some of the common quadrilaterals are –

##### Square

A square is a quadrilateral that has four equal sides and four right angles.

The properties of a square are –

1. A square has four equal sides, i.e. all the sides of a square are equal.
2. A square has four right angles
3. A square has two pairs of parallel sides
4. The diagonals of a square bisect each other
5. The diagonals of a square are perpendicular to each other

Following is the example of a square –

##### Rectangle

A rectangle is a type of quadrilateral that has equal opposite sides and four right angles. The properties of a rectangle are –

1. A rectangle has two pairs of parallel sides
2. A rectangle has four right angles
3. A rectangle has opposite sides of equal lengths
4. The diagonals of a rectangle bisect each other

Following is the example of a rectangle –

##### Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

The properties of a parallelogram are –

1. A parallelogram has opposite sides of equal lengths.
2. The opposite angles of a parallelogram are equal.
3. A parallelogram has two pairs of parallel sides
4. The diagonals of a parallelogram bisect each other

Following is the example of a parallelogram –

##### Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides is parallel.

The properties of a trapezium are –

1. A trapezium has one pair of parallel sides
2. A trapezium has no right angles
3. A trapezium has one pair of opposite sides of equal lengths

Following is the example of a trapezium–

##### Rhombus

A rhombus is a quadrilateral with four equal sides.

The properties of a rhombus are –

1. A rhombus has two pairs of parallel sides
2. The opposite angles in a rhombus are equal.
3. A rhombus has four equal sides. This means that all its sides are equal.
4. The diagonals of a rhombus bisect each other
5. The diagonals of a rhombus are perpendicular each other

Following is the example of a rhombus –

### Classification on the Basis of Size

Objects can also be classified based on their size. Let us consider the following objects –

We can clearly see that the four objects that have been given above are of different sizes. The first object is of the smallest size while the fourth object is the largest of them all. Hence, classification based on sizes is another way in which we can categorise different objects.

### Classification on the Basis of Colour

Objects can also be classified based on their colour. Let us consider the following objects –

We can clearly see that the four objects that have been given above are of different colours.  The first object is of red colour. The colour of the second object is brown. The third object is of green colour while the colour of the fourth object is pink. Hence, classification based on colours is another way in which we can categorise different objects.

### Classification on the Basis of Numbers

Objects can also be classified based on the number of objects. Let us consider the following objects –

The first object is a group of two circles. The second object is a group of three pentagons. The third object is one rectangle. The fourth object is a group of five triangles. Hence, classification based on the number of objects is another way in which we can categorise different objects.

## 3 – Dimensional Shapes

3 Dimensional shapes or 3D shapes are the shapes that have all three dimensions, i.e. length, breadth and height.  These shapes are also called solid shapes. 3 – Dimensional shapes are further classified on the basis of the arrangements of the sides and the angles.  Let us learn about them.

### Cuboid

A 3D shape having six rectangular faces is called a cuboid, ex a matchbox, a brick, a book etc. In other words, it is an extension of a rectangle in a 3D plane.

Below we have a general diagram of a cuboid

### Cube

A cuboid whose length, breadth and height are equal is called a cube. Examples of a cube are sugar cubes, cheese cubes and ice cubes. In other words, it is an extension of a square in a 3D plane.

Below we have a general diagram of a cube

### Cylinder

A cylinder is a solid with two congruent circles joined by a curved surface. Objects such as a circular pillar, a circular pipe, a test tube, a circular storage tank, a measuring jar, a gas cylinder, a circular powder tin etc. are all shapes of a cylinder. A cylinder has a curved lateral surface and two circular faces at its ends.

Below we have a general diagram of a Cylinder

### Cone

A circular cone has a circular base that is connected by a curved surface to its vertex. A cone is called a right circular cone if the line from its vertex to the centre of the base is perpendicular to the base. An ice-cream cone is an example of a cone

Below we have a general diagram of a Cone

### Sphere

A sphere is a solid formed by all those points in space that are at the same distance from a fixed point called the centre.  In other words, it is an extension of a circle in a 3D plane.

Below we have a general diagram of a Sphere

## Classification of Numbers

Now that we have learnt about the classification of objects and figures, let us also discuss some classifications of numbers. The numbers can be classified into different categories. Some of these classifications are –

### Odd Numbers

Odd numbers are numbers that are not divisible by 2, and always leave a remainder 1 when divided by 0. The odd numbers end up in any of the five digits 1, 3, 5, 7 and 9. Therefore we can say that all odd numbers have any of the five digits 1, 3, 5, 7 and 9 at their units place.

### Even Numbers

Even numbers are numbers that are divisible by 2, leaving a remainder 0. The even numbers end up in any of the five digits 0, 2, 4, 6 and 8. Therefore we can say that all even numbers have any of the five digits 0, 2, 4, 6 and 8 at their units place.

### Prime Numbers

The numbers that have only two factors, i.e. 1 and the number itself are called prime numbers. For example, consider the number 5. How many factors does the number 5 have? The factors of 5 are 1 and 5. This means that 5 has only two factors, 1 and the number 5 itself. Hence, it is a prime number.

### Composite Numbers

Composite numbers are numbers with more than two factors. In other words, composite numbers are those natural numbers that have more than 2 factors.

## Key Facts and Summary

1. Classification is the term used to represent the comparison of objects and numbers based on their similarities and differences.
2. 2 Dimensional shapes or 2D shapes are the shapes that have only length and breadth. For example, the shapes such as rectangles and squares are 2-dimensional shapes.
3. A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance from a fixed point in the plane. This fixed distance is called the radius of the circle and the fixed point is called the centre of the circle.
4. A triangle is a polygon that is made up of 3 sides, 3 angles and three vertices
5. A scalene triangle is a triangle in which all the three sides are of unequal lengths. This means that in a scalene triangle, no two sides are equal.
6. A triangle is said to be an isosceles triangle if two of its sides are equal.
7. A triangle is said to be an equilateral triangle if all its sides are equal.
8. A triangle is said to be a right angled triangle if one of the angles of the triangle is a right angle, i.e. 90o.
9. A square is a quadrilateral that has four equal sides and four right angles.
10. A rectangle is a type of quadrilateral that has equal opposite sides and four right angles.
11. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
12. A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
13. A rhombus is a quadrilateral with four equal sides.
14. 3 Dimensional shapes or 3D shapes are the shapes that have all the three dimensions, i.e. length, breadth and height.  These shapes are also called solid shapes.
15. A 3D shape having six rectangular faces is called a cuboid, ex a matchbox, a brick, a book etc. In other words, it is an extension of a rectangle in a 3D plane.
16. A cuboid whose length, breadth and height are equal is called a cube.
17. A cylinder is a solid with two congruent circles joined by a curved surface.
18. A circular cone has a circular base that is connected by a curved surface to its vertex.
19. A sphere is a solid formed by all those points in space that are at the same distance from a fixed point called the centre.
20. Odd numbers are numbers that are not divisible by 2, and always leave a remainder 1 when divided by 0.
21. Even numbers are numbers that are divisible by 2, leaving a remainder 0.
22. The numbers that have only two factors, i.e. 1 and the number itself are called prime numbers.
23. Composite numbers are numbers with more than two factors.