Symmetry And Line Of Symmetry: An Introduction

Symmetry is defined as a balanced and a proportionate similarity which is found in two halves of an object, that is, one-half is the mirror image of the other half. The imaginary line or axis along which you can fold a figure to obtain the symmetrical halves is called the line of symmetry.  In this article, let us discuss the meaning of symmetry in Maths, line of symmetry, types of symmetry with more examples and video tutorial.

Symmetry in Mathematics

In Mathematics, a meaning of symmetry defines that one shape is exactly like the other shape when it is moved, rotated, or flipped. Consider an example, when you are told to cut out a ‘heart’ from a piece of paper, don’t you simply fold the paper, draw one-half of the heart at the fold and cut it out to find that the other half exactly matches the first half? The heart carved out is an example of symmetry.


Symmetry Math definition states that “symmetry is a mirror image”. When an image looks identical to the original image after the shape is being turned or flipped, then it is called symmetry. It exists in patterns. You may have often heard of the term ‘symmetry’ in day to day life. It is a balanced and proportionate similarity found in two halves of an object, that is, one-half is the mirror image of the other half. And a shape that is not symmetrical is referred to as asymmetrical. Symmetric objects are found all around us, in nature, architecture, and art.

Line of Symmetry

The imaginary line or axis along which you fold a figure to obtain the symmetrical halves is called the line of symmetry. It basically divides an object into two mirror-image halves. The line of symmetry can be vertical, horizontal or diagonal. There may be one or more lines of symmetry.

  • 1 line: Figure is symmetrical only about one axis. It may be horizontal or vertical. The word ATOYOTA has one axis of symmetry along the axis passing through Y.


  • 2 lines: Figure is symmetrical only about two lines. The lines may vertical and horizontal lines as viewed in the letters H and X.


  • Infinite lines: Some figures have not one or two, but infinite lines passing through the centre, and the figure is still symmetrical. Example: a circle.



Types of Symmetry

Symmetry may be viewed when you flip, slide or turn an object. There are four types of symmetry that can be observed in various situations, they are:

Translation Symmetry: If the object is translated or moved from one position to another, the same orientation in the forward and backward motion is called translational symmetry. In other words, it is defined as the sliding of an object about an axis. This can be observed clearly from the figure given below, where the shape is moved forward and backward in the same orientation by keeping the fixed axis.

Translation symmetry

Rotational Symmetry: When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry or radial symmetry. Rotational symmetry existed when a shape turned, and the shape is identical to the origin. The angle of rotational symmetry is the smallest angle at which the figure can be rotated to coincide with itself. The order of symmetry is how the object coincides with itself when it is in rotation.

In geometry, many shapes consist of rotational symmetry. For example, the figures such as circle, square, rectangle have rotational symmetry. Rotational symmetry can also be found in nature, for instance, in the petals of a flower.

Below figure shows the rotational symmetry of a square along with the degree of rotation.

Rotational symmetry

Click here for more information about rotational symmetry.

Reflexive Symmetry: Reflection symmetry is a type of symmetry in which one half of the object reflects the other half of the object. It is also called mirror symmetry or line of symmetry. A classic example of reflection symmetry can be observed in nature, as represented in the below figure.

Reflexive symmetry

Read more about reflection symmetry here.

Glide Symmetry: The combination of both translation and reflection transformations is defined as the glide reflection. A glide reflection is commutative in nature. If we change the combination’s order, it will not alter the output of the glide reflection.

Symmetry Shapes

The symmetry of shapes can be identified whether it is a line of symmetry, reflection or rotational based on the appearance of the shape. These can be better understood with the examples given below.

Solved Examples

Example 1: If the figure below follows a reflexive or line of symmetry, then complete the figure.

Symmetry Example Q1


Given that, the figure has a line of symmetry.

That means, the second half (i.e. missing part) of the figure will be exactly the same as the given.

Thus, the complete figure is:

Symmetry Example Q1 Solution

Example 2: Identify the shapes which do not have rotational symmetry from the below figure.

Symmetry Example Q2


As we know, rotational symmetry is a type of symmetry, when we rotate a shape in a particular direction, the resultant shape is exactly the same as the original shape.

Thus, from the given figure (a) and (c) do not have a rotational symmetry.

Frequently Asked Questions on Symmetry – FAQs

What is symmetry with example?

The word symmetry is the most commonly used concept in the study of reflections of mages. It is often referred to as mirror or reflective symmetry; that means a line or plane that can be drawn through an object such that the two halves are mirror images of each other.

What are the 4 types of symmetry?

The four types of symmetry are:
Translation symmetry
Rotational symmetry
Reflection (or reflexive) symmetry
Glide symmetry

What is symmetric in math?

In Mathematics, the meaning of symmetry is that one shape is exactly like the other shape when it is moved, rotated, or flipped.

Why is symmetry in nature?

Symmetry lies at the heart of the laws of nature. We can observe symmetry in many things that exist in this nature. For example, a line of symmetry can be observed in flowers, butterflies and so on.

Can a line of symmetry be parallel?

No, not all lines that divide an object into two congruent halves are lines of symmetry. Yet if we try to be creative and draw the line parallel to a set of sides, the folding does not coincide with the shape’s other side. Thus, the line of symmetry cannot be parallel.

A number of other kinds of symmetric types exist such as the point, translational, glide reflectional, helical, etc. which are beyond the scope of learning at this stage. Know much more about two lines of symmetry and reflection symmetry and also get the detailed solutions to the questions of the NCERT Books for the chapter Symmetry at BYJU’S.


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