# What is Rotational Symmetry ?

In this chapter we will look into the concept of rotational symmetry with properties and examples.

## Rotational symmetry definition?

If an object after a rotation looks the same then the object has rotational symmetry.

For example;
Consider the English letter ” H “

If we rotate the letter ” H ” by 180 degree, we would get the same “H” looking letter.

Observe the above image which is divided into three parts.

Part (A) ⟹ Original letter ” H “

Part (B) ⟹ Image rotated by 90 degree

Part (C) ⟹ Image rotated by 180 degree

Note that Part (A) and (C) looks the same.

So, if we rotate the letter ” H ” by 180 degree, we would get the same looking letter.

Hence, the letter ” H ” has rotational symmetry.

Note:
To have rotational symmetry, the object must look the same before 360 degree full rotation.

Once we rotate the object by 360 degree, we are back to the original image.

### Method of Rotation for Rotational Symmetry

There are two ways to rotate any given object;

(a) Clockwise Rotation
(b) Anti-clockwise Rotation

#### Clockwise Rotation

Given below is the example of clockwise rotation about its center.

Note that in all the rotation, the object doesn’t look the same.

Hence, the given letter ” M ” doesn’t have rotation symmetry.

#### Anti-clockwise Rotation

We will rotate the same letter ” M ” in anti-clockwise direction.

Note:
You can use both the rotation technique for Rotational symmetry as both will yield the same result.

### Angle of Rotation

The angle at which the given object is rotated is called angle of rotation.

Usually in geometry, you will see four types of rotation.

(a) 90 degree rotation – Quarter turn
(b) 180 degree rotation – half turn
(c) 360 degree rotation – full turn

Important Note:
The object is said to be in rotational symmetry if before 360 degree full rotation it looks the same at-least once.

## Rotational Symmetry – Solved Problems

(01) Check which of the following images has rotational symmetry

Solution
In each of the above image we will check if rotating the object results in same image.

Object (A)
The given object is a rectangle.

Let us rotate the object clockwise by 90 degree each.

Observe that the rectangle is rotated clockwise by 90 degree around central point.

Here image (i) and (iii) and (v) looks the same.

Since the object looks same before full 360 degree rotation, the given rectangle has rotational symmetry.

Object (B)
The given object is a triangle.

Let’s rotate the triangle by 90 degree clockwise to check if it has rotational symmetry.

Observe that in above rotation, we don’t get same image until the object is fully rotated to 360 degree.

Hence, the given triangle is not in rotational symmetry.

Object (C)
The given object in English Letter ” S “

Rotate the letter by 90 degree clockwise around the central point.

Note that image (i), (iii) and (v) are same.

This means that the object repeat itself before 360 degree rotation. Hence, the given object has rotational symmetry.

Object (D)
The given object is English letter “Z”

Rotate the object by 90 degree clockwise.

Note that image (i), (iii) and (v) are same.

This means that the image repeats itself before 360 degree full rotation.

Hence, the given object has rotational symmetry.