Symmetry

What is Symmetry?

Symmetry: Definition

An object has a symmetry if it can be divided into two identical pieces.

When an object has a symmetry, we say that it is symmetrical.

If an object does not have a symmetry, we say that the object is asymmetrical.

The concept of symmetry is commonly found in geometry.

The following images are symmetrical.

Symmetry in Geometry is explained with two images. One is that of a heart and the other is a polygon.


Line of Symmetry

The line of symmetry is a line that divides an object into two identical pieces.

Here, we have a star and we can fold it into two equal halves.

Symmetry in Geometry is explained using a star. There is a line that divides the star into two, equal parts.

If you fold a figure in half along its line of symmetry, you will notice that both the halves match each other exactly.

This line of symmetry is called the axis of symmetry.

The line of symmetry is vertical if it cuts the shape from top to bottom and vice-versa.

Types of Symmetry in Math

The line of symmetry is horizontal if it cuts the shape from right to left and vice-versa.

Types of Symmetry in Math

Sometimes, we can split a shape across the corners to form two identical halves.

In such a case, the line of symmetry is diagonal.

Types of Symmetry in Math

Now its time to explore the symmetrical shapes visually.

Mark one line of symmetry by selecting the points


Shapes with more than one Line of Symmetry

Some symmetrical figures have a single line of symmetry while others have more than one.

Consider this triangle.

It has only one line of symmetry.

If you try to divide it any other way, the parts will be asymmetrical. 

Lines of Symmetry is explained using a triangle.

In comparison to the above triangle, the one shown below has 3 lines of symmetry. 

Lines of Symmetry is explained using an equilateral triangle.

 

Number of lines of symmetry Examples of figures
No line of symmetry Scalene triangle
Exactly one line of symmetry Isosceles triangle
Exactly two lines of symmetry Rectangle
Exactly three lines of symmetry Equilateral triangle

Making a Symmetrical Figure: Ink String Patterns

Let us do a simple activity.

Dip short lengths of strings in different coloured paints or inks.

Now, place these strings on half of a paper.

Fold the paper and press the two halves.

Pull out the strings from the paper.

Is the resulting figure symmetric?

If yes, what is the line of symmetry?

Try out more such patterns!

 
Thinking out of the box
Think Tank
 
1.  The designs on the playing cards have a line of symmetry. Can you identify them for the following cards?
 

Playing cards are an example of symmetrical objects.

2. Can you list the letters in the English alphabet from A to Z that have no lines of symmetry?

Reflection and Rotational Symmetry

Reflection Symmetry

When one half of an object is exactly the same as the other half of the object, it has reflection symmetry.

It can be observed that the two halves are reflections of each other.

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

Examples of Symmetry - A woman's face is shown as an example for reflection symmetry.

Reflection of mountains and trees in clear water is an example of reflection symmetry.

Rotational Symmetry

Look at this starfish.

Examples of Symmetry in Real Life - A starfish is shown as an example for rotational symmetry.

This starfish has 5 lines of symmetry that meet at a point P.

We can rotate the starfish about the point P.

Examples of Symmetry in Real Life - A starfish is shown as an example for rotational symmetry.

If you turn or rotate the starfish about the point P, it will still look the same from all directions.

This kind of symmetry is called rotational symmetry.

The famous ferris wheel, the London Eye, is an example of rotational symmetry.

You can find many objects in real life that have rotational symmetry like wheels, windmills, road-signs, ceiling fans and so on.

Let's play with some figures having reflection and rotational symmetry.

Symmetry is everywhere around us.

Almost all plants, animals, and even humans are symmetric.


Point Symmetry

An object has a point symmetry if every part of the object has a matching part.

Many letters of the English alphabet have a point symmetry.

The point O is the central point and the matching parts are in opposite directions.

Symmetrical Objects in Daily Life - Some alphabets are shown as examples of point symmetry.

If an object looks the same when you turn it upside down, then it is said to have a point symmetry.

The shape and the matching parts must be in opposite directions.

 
important notes to remember
Important Notes
  1. All regular polygons are symmetrical in shape. The number of lines of symmetry is the same as the number of its sides.
  2. An object and its image are symmetrical with reference to its mirror line.
  3. If a figure has a rotational symmetry of \(180^{\circ}\), then it has a point symmetry.

Examples of Symmetry

Symmetry in Real Life

Example 1

 

 

Symmetry is often found in nature.

One common example is a butterfly.

When a butterfly folds its wings, it is easy to observe symmetry in its wings. 

The wings of most of the butterflies are identical on the left and right sides.

Symmetrical Objects in Real Life - A butterfly is an example of a symmetrical object found in nature.

Example 2

 

 

Symmetry is one of the most prevalent themes in art, architecture, and design.

Do you know why these architectural marvels of India are beautiful?

  Examples of Symmetry in Real Life - Architectural monuments such as the Taj Mahal are examples of symmetrical patterns.Examples of Symmetry in Real Life - Some architectural monuments depict symmetrical patterns.

This is because of symmetry! 

Thus, the geometry and the symmetry of the structure gives uniqueness to these monuments.

Example 3

 

 

Observe this beautiful figure.

Symmetric shapes and patterns appear more beautiful than non-symmetric ones.

 Symmetry in our daily life - A snowflake is symmetrical.

This is Koch’s Snowflake.

It has a symmetrical pattern.

You can find many similar, beautiful objects that have symmetry in real life.


Symmetry: Fun Facts

A kaleidoscope has mirrors inside it that produces images that have multiple lines of symmetry.

The angle between the mirror decides the number of lines of symmetry.

 Symmetry in our daily life - Kaleidoscopes produce images that have multiple lines of symmetry.

We may have observed several symmetrical objects in our daily life like rangolis or kolams. 

The striking aspect of symmetry can be observed in rangoli designs.

These designs are famous in India for their unique and symmetrical patterns.

They depict the colourful science of symmetry.

 Symmetry in our daily life - Rangolis include symmetrical patterns in them.


Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is symmetry in math?

When an object is exactly the same when you turn it or flip it, that object has a symmetry.

Symmetrical objects are of the same size and shape.

2. What are the 4 types of symmetry?

The main four types of symmetry are rotational symmetry, reflectional symmetry, point symmetry and translational symmetry.

3. What is symmetry? Explain with an example.

Symmetry Definition: When an object is exactly the same when you turn it or flip it, that object has a symmetry.

Symmetrical objects are of the same size and shape.

Nature has plenty of objects having symmetry. 

Some common examples for symmetry are shown below.

Symmetry in Math - Flowers, leaves, and bees are some examples of objects found in nature that have symmetrical patterns.

4. What is a symmetric pattern?

All patterns having a symmetry are called symmetric patterns.

Leaves of plants have various patterns and shapes. Most of these leaves depict symmetric patterns.

For example, the two sides of a leaf look similar with respect to their shape, size and structure.

 
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