Rotational Symmetry: Angle, Order & Examples

The concept of Rotational symmetry explains that when you rotate an object on its own axis, the shape of the object remains the same. Various figures are symmetrical when they are rotated 180 degrees or with some angles, clockwise or anticlockwise.

• Rotational symmetry is also known as radial symmetry.
• With this symmetry, the object remains the same even after partial rotation.
• The rotation of such figures is direct isometries, which means these figures preserve orientation.
• We know that the word symmetry is a combination of two words ‘sync + meter.
• This is why there should be at least two identical orders to have symmetry. 
• We can find symmetry all around us, in nature, in architecture, and in art. 
• It may be explored when we flip, slide or turn an object. 
• There may be different types of symmetry, such as reflection, translational and rotational. 
• Examples of rotational symmetry include shapes such as squares, circles, hexagons, etc. 

Key Terms: Rotational Symmetry, Rotation, Order of Rotational Symmetry, Symmetry, Hexagon, Star, Reflection, Angle of Symmetry, Scalene Triangle, Center of Rotation, Circle, Squares

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What is Rotational Symmetry?

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The theory of Rotational Symmetry of shape states that an object looks the same when rotated on its axis. 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. 

• Rotational Symmetry is a balanced and proportionate similarity found in two halves of an object.
• In this method, one-half is the mirror image of the other half.
• The shapes which are not symmetrical are referred to as asymmetrical.
• We may have heard of the term ‘symmetry’ in our day-to-day life.
• When the images are rotated around the origin, then they belong to a class of orthogonal group.
• However, a scalene triangle does not have symmetry if rotated as the shape is asymmetrical.

There are many shapes in the field of geometry that follow the concept of rotational symmetry, which are as follows:

• Square
• Hexagon
• Circle
• Rectangle
• Rhombus

Rotational Symmetry

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Center of Rotation

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The center of rotation is defined as the rotation of an object around a fixed point. The object should follow the rotational symmetry. Rotation is done either clockwise or counterclockwise.

• The rotation is symmetry in m-dimensional Euclidean group.
• The rotated shape is identical to the origin.

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Angle of Rotational Symmetry

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The angle of rotational symmetry is defined as the smallest angle in which a shape is turned during rotation. After flipping, the shape of the object remains intact. When an object is rotated along rotational symmetry, then it will complete a rotation of 360 degrees.

Some of the angles of rotational symmetry include:

• When an object is rotated at an angle of 90 degrees, then it is called the quarter turn.
• When you rotate an object at an angle of 180 degrees, then it is called a half turn.
• When an object is rotated at an angle of 360 degrees, then it is called the complete turn.

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Order of Rotational Symmetry

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Order of rotational symmetry is the number of times any shape or an object is rotated. It still looks similar to us as it was before the rotation. The rotational symmetry order of a regular hexagon is equivalent to the number of sides a polygon has.

• We can find the order of symmetry by determining the smallest angle we can rotate any shape.
• It will look the same as the original figure.
• With respect to rotational symmetry, the center of any shape or object is the point around which rotation appears.

Various order of rotational symmetry are as follows:

• 80° = Order 2
• 120° = Order 3,
• 90° = Order 4.

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Rotational Symmetry Letters

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English alphabets consists of various letters that follow the properties of rotational symmetry. They maintain their symmetry even after rotation in clockwise and anticlockwise direction at 180 degrees.

• For your knowledge, there are many rotational symmetry letters in the English alphabet, including Z, H, S, N, and O.

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Things to Remember

• The rotational symmetry includes the rotation of an object on its own axis, and the shape of the object looks the same.
• The order of rotational symmetry can be defined by the smallest angle we can rotate any shape so that it looks the same.
• When some alphabets are rotated 180 degrees clockwise or anticlockwise, the letters appear to be the same.
• The swastika symbol and roundabout symbol you have seen in your daily life exhibit the principle of rotational symmetry.
• It is a subgroup of the Euclidean group.
• The physical system will follow the angular momentum of the law.

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