Rotational Symmetry

Rotational Symmetry Meaning

Rotational Symmetry of shape states that an object looks the same when it is rotated on its axis. Most of the geometrical shapes seem to appear as a symmetry when they are rotated clockwise, anticlockwise or rotated with some angle such as 180°,360°, etc. Some of the examples of geometrical shapes that appear as symmetry are square, hexagon and circle. A scalene triangle does not appear to be symmetrical when rotated. Hence, it is asymmetrical in shape.

There should be at least two similar orders to have symmetry as the word symmetry is a combination of two words ‘sync+metry’. The objects which do not appear to be symmetrical when you flip, slide, or turn are considered as asymmetrical in shape. Symmetry is found all around us, in nature, in architecture, and in art. There are various types of symmetry. These are,

  • Reflection Symmetry 

  • Translational Symmetry 

  • Rotational Symmetry 


What is Rotational Symmetry?

Any figure or shape that rotated around a center point and looks exactly similar as it was before the rotation, is said to have rotational symmetry. Some shapes which have rotational symmetry are squares, circles, hexagons etc.

In Geometry, many shapes have rotational symmetry. These are:

  • Equilateral triangles

  • Squares

  • Rectangles

  • Circles

  • Regular Polygons


How to Determine The Order Of Rotational Symmetry Of Any Shape?

The order of rotational symmetry is the number of times any shape or an object is rotated and still looks similar as it was before the rotation.

The order of rotational symmetry of a regular hexagon is equivalent to the number of sides a polygon has.

The order of rotational symmetry can also be found by determining the smallest angle you can rotate any shape so that it looks the same as the original figure.  

80° = Order 2

120° = Order 3,

90° = Order 4.

The product of the angle and the order will be equal to 360°.


Center of An Object

The center of any shape or object with the rotational symmetry is the point around which rotation appears. For example, if a person spins the basketball on the tip of his finger, then the tip of his finger will be considered as rotational symmetry. If any object has a rotational symmetry then the center of an object will also be its center of mass.


Order of Rotational Symmetry

The number of times any shape or an object that can be rotated and yet looks similar as it was before the rotation, is known as the order of rotational symmetry. For example, a star can be rotated 5 times along its tip and looks similar each time. Hence, its order of rotational symmetry of the star is 5.

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If we examine the order of rotational symmetry for a regular hexagon  then we will find that it is equal to 6. A regular hexagon has 6 equal sides and can be rotated with an angle of 60 degrees.


Angle of Rotational Symmetry

The angle of rotation is the smallest angle a shape is turned or flipped to make it look similar as its original shape. 

  • A quarter turn means a rotation of 90°

  • A half- turn indicates a rotation of 180°

  • A complete turn indicates a rotation of 360°

  • An object is considered as a  rotational symmetry if it string along more than once during a complete rotation, i.e.360°


Rotational Symmetry Letter

There are various English alphabets that have rotational symmetry when they are rotated clockwise or anticlockwise about an axis. Some of the English alphabets which have rotational symmetry are: Z, H, S, N and O.These alphabets will exactly look similar as original when it will be rotated 180 degrees clockwise or anticlockwise


Some of the  examples of rotational symmetry are given below:

Rotational Symmetry Examples

Name

Shape

Rotational Symmetry Order Number

Centre Of Rotation


Parallelogram

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2

Intersection of diagonals

Regular polygon with n sides

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n

Intersection of diagonals

Rhombus

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2

Intersection of diagonals

Circle

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Unlimited

Centre of circles

Trapezium

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None


Kite

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None



Solved Examples

  1. Which of the following figures have rotational symmetry of more than order 1?

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Solution: 

In the above figure, a,b,d,e and f have rotational symmetry of more than order 1.

Figure (a) has rotational symmetry of order 4, figures (b) and (e) has rotational symmetry of order 3, figure (d) has rotational symmetry of order 2, and figure (f) has rotational symmetry of order 4.

  1. Determine the order of rotational symmetry of a square and the angles of such rotation.

If the square is rotated either by 90°,180°,270° or by 360° then the shape of the square will look exactly similar to its original shape. Hence the square has a rotational symmetry of order 4.

  1. Determine the order of rotational symmetry of a rhombus and the angles of such rotation.

If the square is rotated  either by,180,or by 360, then the shape of the rhombus will look exactly similar to its original shape. Hence the rhombus has rotational symmetry of order 2.


Quiz Time

  1. Which of the figures given below does not have a line of symmetry but has a rotational symmetry

  1. H

  2. I

  3. Z

  4. X


2. Determine the smallest angle of rotation that maps the image to itself.

  1. 72

  2. 180

  3. 144

  4. 45

Facts

  • The World’s largest Ferris wheel London eye has rotational symmetry of order 32.

FAQ (Frequently Asked Questions)

1. Explain Line Symmetry, Reflective Symmetry, And Rotational Symmetry.

Line Symmetry- Shapes or patterns that have different types of symmetry, depending on the number of times any shape can be folded in half and still remains similar on both sides. In other words, we can say that the line that divides any figure, shape or any image in similar halves then that figure is said to have line symmetry.

Reflective Symmetry- Reflective symmetry is when a particular shape of pattern is reflected in a line of symmetry. The reflected shape will be similar to the original, the similar size and same distance from the mirror line.

Rotational Symmetry- When any shape or pattern rotates or turns around a central point and remains the same then it is said to have rotational symmetry. For example, if we say that shape has rotational symmetry of order X , this implies that shape can be turned around a central point and still remains the same X times.

2. Can We State That A Circle And Trapezium Have Rotational Symmetry?

A trapezium has one pair of parallel sides. Some trapeziums that include one line of symmetry. Such trapezium is known as isosceles trapezium as they have two sides which are equally as similar to isosceles triangles. A trapezium has rotational symmetry of order 1.

The order of rotational symmetry in terms of a circle refers to the number of times a circle can be adjusted when experimenting with a rotation of 360 degrees. A circle has a rotational symmetry of order that is infinite.