Symmetry

Symmetry in Maths

Look at both the figures given below. What do you see in figure 1 if a line is drawn exactly from the center of the figure it is exactly the same on both sides. That is we draw it on a paper and fold from the center line it will exactly fit each other. The centerline will divide the figure into two identical halves. Such shapes are called symmetrical shapes and the centerline is called the line of symmetry. Here on this page, we will learn the symmetry in maths, line of symmetry, types of symmetry, and its various examples.

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What is Symmetry?

When a shape is divided from the center, it is divided into two identical halves. Or you can say when a figure is divided into two halves one is the mirror image of the other. Consider the below shape of a butterfly, if it is folded from the exact center, it will be identical on both sides. Such shapes are called symmetrical shapes. And the shapes which are not identical on both sides of the centerline are called asymmetrical shapes. The below shape is an example of asymmetrical shape.

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Examples of symmetrical shapes are found all around us, like a flower, a butterfly, etc.

Line of Symmetry

The figure is divided along the center line which is an imaginary line this line is called the axis of symmetry or the line of symmetry. The line of symmetry can be horizontal, vertical, or diagonal. There can be one, two, or more lines of symmetry. From the below figure the rectangle is divided horizontally and vertically. The dashed line is called the line of symmetry. There can be two lines of symmetry in a rectangle.

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• Example of a figure having one line of symmetry

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• Example of a figure having two lines of symmetry.

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• Some figures have an infinite number of lines of symmetry. The best example is a circle with an infinite number of lines of symmetry.

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 Real-life examples of symmetry

• Reflection of trees in clear water.

• Reflection of mountains in a lake.

• Butterflies are identical on both sides.

• Some human faces are identical on the left and right sides.

• People can also have an identical mustache on both sides.

Line of Symmetry for Regular Polygons

Equilateral Triangle

• All three sides of an equilateral triangle are equal.

• It has three lines of symmetry.

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Square 

• All four sides are equal.

• It has four lines of symmetry.

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Regular Pentagon

• A regular pentagon has five equal sides.

• So it has five lines of symmetry.

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Regular Hexagon

• A regular pentagon has six equal sides.

• So it has six lines of symmetry.

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Regular Heptagon

• A regular heptagon has seven equal sides.

• So it has seven lines of symmetry.

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Regular Octagon

• A regular octagon has eight equal sides.

• So it has eight lines of symmetry.

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So we can say that a regular polygon has as many lines of symmetry as the number of sides it has.

Types of Symmetry

Symmetry in Maths is divided into two types of symmetry.

 Types of symmetry are as follows:

• Reflection symmetry

• Rotational Symmetry

Reflection Symmetry: when a figure is divided into two identical halves such that one is a mirror image of the other then it is called reflection symmetry. It is the simplest type of symmetry often referred to as line symmetry or mirror symmetry.

Example:

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Rotational Symmetry:  

A figure that looks exactly the same after rotation as it was before rotation is called rotational symmetry. When an object rotates, its size and shape do not change. Rotation can be clockwise or anticlockwise.

The fixed point about which the object is rotated is called the center of rotation.

Example :

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Order of Rotational Symmetry

The number of distinct orientations in which the shape looks the same as the original is called its order of rotational symmetry.

A full turn means a rotation of about 3600. In a full turn, there are mainly 4 rotational positions

• Rotation through 900, which is called a quarter turn.

• Rotation through 1800, which is called half turn.

• Rotation through 2700, which is called a three - fourth turn.

• Rotation through 3600, which is called a full turn.

Solved Examples

Example 1: Identify the symmetrical shapes from the below figure.

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Solution: Figure a, c and d are the symmetrical shapes because the line of symmetry divides the figure into two identical halves.

Figure b and e are asymmetrical shapes because the line of symmetry does not divide the figure into two identical halves.  

Example 2: Draw the line of symmetry for the following figures.

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 solution:

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Quiz Time

1. How Many Lines of Symmetry a Parallelogram have?

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2. How Many Lines of Symmetry does the Below Shape have?

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