In this chapter we will learn about the Rotational symmetry order with solved examples.

Let us first review the basics of rotational symmetry.

## What is Rotational Symmetry ?

An object is said to be in rotational symmetry when its image repeat itself after rotation.

For example, consider the English letter ” N “

Let us rotate the digit by 90 degrees in clockwise direction.

Note that after rotation the image (i), (iii) and (v) repeats itself.

Since the image repeat itself before full 360 degree rotation, the object is in Rotational symmetry.

## Order of Rotational Symmetry

The Rotational symmetry order tells us the number of times the image repeat itself after full rotation.

Let us understand different order of rotational symmetry.

(i) Rotational symmetry of Order 1

It tells that the image get repeated once after full rotation.

Note that the English Alphabet ” A ” is repeated only at 360 degree full rotation.

Hence, the order of rotational symmetry is 1.

The object with rotational symmetry 1 is not considered to have rotational symmetry because rotating object by 360 degree results same original image.

### (ii) Rotational symmetry of order 2

It says that the image is repeated twice after full 360 degree rotation.

Observe that the original image ” Z ” is repeated twice in part (iii) and part (v).

### (iii) Rotational symmetry of order 3

It means that the object’s image is repeated three times after full 360 degree rotation.

For example, consider the below equilateral triangle.

Observe the above image.

Here we are rotating the equilateral triangle by 120 degree.

Here (i) is the original image and (ii), (iii) & (iv) are the rotated image.

Note that the image is repeating 3 times after full rotation. Hence the image has rotational symmetry of order 3.

### (iv) Rotational symmetry of order 4

If the image repeat itself 4 times after 360 degree rotation, then the object has rotational symmetry of order 4.

Given below is the image of square which is rotated by 90 degree in clockwise direction.

Here;

Image (i) is the original object.

(ii), (iii), (iv) and (v) are the images after rotation which is similar to original image.

We get the same image 4 times after 360 degree rotation. Hence, the order if rotation symmetry is 4.

## Order of Rotational symmetry formula

For any object if you know the angle of rotation at which we get the similar image, then order of rotational symmetry can be calculated as;

**Order of rotational symmetry** = **360 / angle of rotation**

**For example;**

In a square, if we rotate the object by 90 degree, we get the same image.

So, angle of rotation = 90 degree

The order of symmetry can be calculated as;

Order of rotational symmetry = 360 / angle of rotation

Order of rotational symmetry

⟹ 360 / 90

⟹ 4

Hence, the square has 4 order of rotational symmetry.

It means that after 360 degree rotation, the image of original square is repeated 4 times.

**Example 02**

Consider the Pentagon as an example.

If we rotate the pentagon by 72 degree, we will get the same image.

So, angle of rotation = 72 degree

The order of symmetry can be calculated as;

Order of rotational symmetry = 360 / angle of rotation

Order of rotational symmetry

⟹ 360 / 72

⟹ 5

Hence, the order of rotation symmetry for pentagon 5.

It means that if we rotate the pentagon to 360 degree, we get the 5 similar images in between.