Rotational Symmetry

When you rotate a star-shaped figure what do you notice? The shape of a star, when seen from any angle, seems the same.  Have you ever given a thought to the reason behind this? A star is an example of rotational symmetry. The chapter will let you delve into the idea behind this special type of symmetry.

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Rotational Symmetry

It is not necessary that an object must always have a line of symmetry to prove its symmetrical attribute. There may be some objects which show symmetry despite the absence of the line of symmetry. Such symmetrical objects show rotational symmetry.

Rotation means circular movement of an object around a fixed axis. A clock is the most common and regularly-sighted example of rotation. The needles of the clock move around a fixed point to tell us time. The movement followed by them is called clockwise rotation.

If any object follows rotation in a direction opposite to clock’s movement then such rotation is termed as anti-clockwise. Other common examples of rotation are blades of ceiling fan and wheels of a bicycle. This where rotational symmetry finds its origin.

Also known as radial symmetry, rotational symmetry is that characteristic in which the object looks same despite rotation of it. When an object rotates around a fixed axis if its appearance of size and shape does not change then the object is supposed to be rotationally symmetrical. The recycling icon is also an example of rotational symmetry. Have you ever noticed something peculiar about the recycle icon?

Rotational symmetry


You look at it from any direction, or orientation, the shape is unchanged! This is what rotational symmetry does to the figure. In rotational symmetry, despite the circular movement the shape and size of the object or the figure look the same. In the recycling icon, the arrows of the icon appear to be circularly moving which suggests the rotational concept of recycling. When this icon is rotated to any angle, after every halt it shall look the same.

Browse more Topics under Symmetry

Attributes of Rotational Symmetry

Now, let’s consider a paper windmill to understand the attributes of rotational symmetry in a better way. A paper windmill is the most joyous toy for every child. Its rotational movement amazes every child, but its unchanged appearance despite continuous rotation adds to the whole amazement.

Rotational symmetry

[source: powersupply]

A paper windmill is a perfect example of symmetry. Now, you may be wondering that in a paper windmill there is no line of symmetry, neither can it be folded. Then what kind of symmetry does it show? Paper windmill shows rotational symmetry. It is fixed to a stick by a pin.

This pin is its center of rotation as its blades rotate around the pin. When we forcefully rotate it to an angle of 90° then we see despite the movement there is no change in its appearance. On further movement, the shape of the paper windmill is the same as the original. The figure below illustrates this more accurately!

Rotational symmetry

The angle at which this paper mill is turned is called the angle of rotation. Here the angle of turn is 90°. A full turn of rotation measures 360°. So for taking one full turn, our paper windmill is moved four times. Do you notice any change after a full turn? The answer shall be a “No”.

The reason for this is that the paper windmill shows rotational symmetry. The number of turns given to an object to come back into its original position gives the order of rotational symmetry for that object. In the case of Paper windmill, rotational symmetry is of order 4.

Some More Examples

A square and a star also show rotational symmetry. Let’s consider a star-shaped starfish:

Rotational symmetry

A starfish is moved at an angle of 72° for one turn. So to give it a full turn of 360° we need to give it 5 turns. This means that any star-shaped object will show the rotational symmetry of order 5. Likewise in a square with O as the center of rotation, we need to give a turn of 90° for one turn.

It takes 4 turns to bring the square to its original form. The square, therefore, shows a rotational symmetry of order 4. Things to be noted in object showing radial symmetry

  1. The center of rotation is the center of the object.
  2. The angle of turn depends on the shape of the object.
  3. Whatever the angle of turn, for a full rotation the angle is 360°.
  4. The order of rotation shall be the number of turns taken to complete one full turn.

If on every turn, no matter what the angle, the shape of the object looks like its original then the object shows radial symmetry.

Solved Examples for You

Question 1: From the following figures, identify the figures which have rotational symmetry of more than order 1:

Rotational symmetry

  1. b,c,e,f
  2. a,b,d,e,f
  3. a,d,e,f
  4. a,c,d,f

Answer : B) In the figures, a,b,d,e, and f the rotational symmetry is more than order 1. Figure (a) has rotational symmetry of order 4. Figures (b) and (e) show a radial symmetry of order 3 while the figure (d) has radial symmetry of order 2. Figure f) shows rotational symmetry of order 4.

Question 2: Explain rotational symmetry with examples?

Answer: Rotational symmetry can be found in many shapes like squares, rectangles, circles, and all regular polygons. To understand rotational symmetry, choose an object and do its rotation to 180 degrees around its centre. The object will be said to have rotational symmetry if at any point, the appearance of the object is exactly like it had before the rotation.

Question 3: What are the various types of symmetry?

Answer: The various types of symmetry are: reflection, glide reflection, rotation, and translation.

Question 4: Can we say that a trapezium has rotational symmetry?

Answer: A trapezium involves one pair of parallel sides. There are some trapeziums that have one line of symmetry. Such trapeziums are known as isosceles trapeziums as they have two equal sides in similarity to isosceles triangles. A trapezium can be said to have rotational symmetry of order one.

Question 5: Can we say that a circle has rotational symmetry?

Answer: The order of rotational symmetry with regards to a circle refers to the number of times a circle fits on to itself when undertaking a rotation of 360 degrees. A circle is associated with an order of rotational symmetry that is infinite.

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