Symmetry

Symmetry is the phenomenon of dividing a geometric shape into the same halves. We can also call symmetry a balanced and proportionate similarity between two halves of an object. Symmetry is an important mathematical concept related to pictorial represented objects and figures.

In this article, we will discuss symmetry in detail and have a look at some solved figure-based examples that will help in understanding the concept of symmetry more clearly.

What is Symmetry in Math?

Symmetry in mathematics is a phenomenon when a geometrical shape or object can be split into two or more equal parts along the symmetrical axis and each of the parts obtained would be identical to each other. Symmetry can be understood as the mirror image of the other portion of the image. The similar parts of the object/ figure are mirror images of each other. Symmetrical figures have an which is present and divides the figure into parts that are equal and identical to each other in all aspects.

Symmetry in mathematics refers to a balance or similarity in shape, size, or arrangement on both sides of a dividing line or point. It’s a fundamental concept used to analyze patterns and structures in mathematics.

Symmetry Definition

Symmetry of any shape is defined if we cut any shpe into two or more identical parts. For example cutting a square along its any diagonal results in the formation of two isosceles triangle that are similar to each other. Symmetry of various shapes is added in the table below,

Number of Lines of Symmetry	Examples
No line of Symmetry	Scalene Triangle
One Line of Symmetry	Isosceles Triangle
Two Line of Symmetry	Rectangle
Three Line of Symmetry	Equilateral Triangle
Infinite Line of Symmetry	Circle

Symmetry Meaning

Symmetry refers to the concept of balance and harmony. It signifies a visual or conceptual equivalence on either side of an axis or a center point. This idea of proportion and balance can be observed in various aspects of our natural world, art, and mathematics.

Symmetry plays a pivotal role in architectural design, where buildings are often crafted to achieve an aesthetic equilibrium. In nature, it can be seen in the graceful patterns of a butterfly’s wings or the radial symmetry of a sunflower.

Symmetrical Figures

Some of the common examples of symmetry are listed below in the figure. In the below attached figure, we can find that each figure has some imaginary lines which divide the figure into two halves and these two halves are the same in shape and size. These imaginary Lines are called Lines of Symmetry or Axis of Symmetry. Various figures that show symmetry are added in the image below,

An object or figure is said to be symmetrical if it’s found to be similar to the other half of the object or figure. For dividing the object, into two halves an axis of symmetry is taken into consideration.

Properties of Symmetrical Figures

The properties of symmetrical figures are mentioned below:

• Symmetrical figures have a line of symmetry passing through their center.
• Each part obtained after the symmetrical axis/ line of symmetry passing through the figure is identical and mirror image of each other.
• There can be more than one line of symmetry of figures and each lin of symmetry would be dividing the figure into equal mirror parts.

Axis of Symmetry

The symmetrical axis also called the line of symmetry which divides the figure into two or more equal parts. Line of symmetry can be vertical, horizontal or diagonal in nature. The symmetrical axis divides the figure into parts that are equal and identical to each other. We can have more than one axis of symmetry for figures and objects. Symmetry axis lines are thus imaginary lines that divide an object into two or more equal and mirror-image halves, ensuring balanced and identical reflection. They are essential in geometry and design for creating harmonious and visually appealing shapes.

Below is the representation of axis of symmetry:

Lines of Symmetry

The lines of symmetry are the imaginary lines along which a geometrical figure is symmetric in nature. We can classify the lines of symmetry in two types. First one is based on the nature of lines of symmetry which can be further classified as horizontal, vertical or diagonal in nature. Second type of line of symmetry is based on the number of lines of symmetry based on which which can classify line of symmetry as

• Horizontal Line of Symmetry
• Vertical Line of Symmetry
• Diagonal Line of Symmetry

Horizontal Line of symmetry

Figures with a horizontal line passing through them and dividing them into identical parts are said to have a horizontal line of symmetry passing through them.

Vertical Line of symmetry

Figures with a vertical line passing through them and dividing them into identical parts are said to have a vertical line of symmetry passing through them.

Diagonal Line of symmetry

Figures with a diagonal line passing through them and dividing them into two or more identical parts are said to have a diagonal line of symmetry passing through them.

Other line of symmetry are,

• One line of symmetry
• Two lines of symmetry
• Infinite lines of symmetry

Let’s learn about them in brief.

One Line of Symmetry

Figures with a single line passing through them and dividing them into two identical parts are said to have a onel line of symmetry passing through them. For Example

Two Line of Symmetry

Figures with two lines passing through them and dividing them into two or more identical parts are said to have two line of symmetry passing through them.

Three Line of Symmetry

Figures with three line passing through them and dividing them into identical parts are said to have a diagonal line of symmetry passing through them.

Four Line of Symmetry

Figures with four line passing through them and dividing them into identical parts are said to have four line of symmetry passing through them.

Infinite Line of Symmetry

Figures with infinite lines passing through them and dividing them into many identical parts are said to have infinite line of symmetry passing through them.

Types of Symmetry

Symmetry can be classified into several distinct types, each playing a significant role. The different types of symmetry are fundamental concepts that help us understand and describe patterns, shapes, and structures in the world around us. Below are the different types of symmetry in mathematics:

• Rotational Symmetry
• Translational Symmetry
• Reflexive Symmetry
• Glide Symmetry
• Point Symmetry

Rotational Symmetry

Rotational symmetry refers to the property of a shape or object to remain unchanged under a rotation about a specific point. This type of symmetry often involves regular polygons, such as a circle, where any degree of rotation leaves the shape looking the same. It’s a fundamental concept in geometry and design.

Translational Symmetry

Translational symmetry relates to the ability of a figure or design to retain its form and appearance when shifted along a straight line in a specific direction. This type of symmetry is frequently observed in patterns, such as a repeating wallpaper design, where moving it left or right, up or down, keeps the pattern consistent.

Reflexive Symmetry

Reflexive symmetry pertains to mirror or reflective symmetry, where an object or shape is symmetrical across a particular axis or line. It means that if you were to fold the object in half over this axis, both sides would perfectly match. Mirroring a shape is a common technique in art and design.

Glide Symmetry

Glide symmetry combines both translational and reflective elements. It involves the ability of an object to be reflected across an axis and then translated (shifted) parallel to the axis. The resulting figure retains its appearance. This type of symmetry is less common but adds complexity to patterns and designs.

Point Symmetry

Point symmetry, often known as central symmetry, occurs when an object looks the same when rotated by 180 degrees around a central point. The figure retains its shape and orientation after this rotation. This type of symmetry is found in various natural and man-made objects, including snowflakes and stars.

Symmetry in Real Life

In real life symmetry can be discussed in various shapes, such as Triangles, Quadrilateral, Pentagon, and Alphabetical Letters, Faces, etc. Let’s learn about them in detail.

Symmetrical Shapes

There are several figures which can be said to symmetrical. The example of the them would be like a cube if the symmetrical axis passes through the middle of the cube. Sphere, circle are also considered to be symmetrical figure with line of symmetry passing through their center. Symmetry in Shapes involves ensuring that the left and right halves of a shape is evenly balanced, creating a pleasing and harmonious appearance.

Symmetry in Triangle

• Triangle symmetry refers to a property where a triangle can be divided into two or more identical parts by a specific axis or center point.
• It occurs when a triangle’s sides, angles, or vertices exhibit balanced characteristics, creating a sense of harmony and equilibrium within the shape.

Symmetry in Quadrilateral

• Quadrilateral symmetry denotes the balanced arrangement of a four-sided polygon, where dividing it along certain lines or points results in identical sections.
• Whether through vertical, horizontal, or diagonal symmetry, a quadrilateral’s symmetrical properties contribute to its aesthetic and geometric appeal.

Symmetry in Pentagon

• Pentagon symmetry involves the five-sided polygon’s capacity to be divided in ways that generate mirror-image segments or angles, showcasing a pleasing and balanced structure.
• From pentagon rotation symmetry to line or point symmetry, these elements contribute to the captivating visual allure of this geometric shape.

Symmetry in Letters

Letter symmetry refers to the visual balance and mirror-like quality of letters and characters in written or printed text. It is a concept that is often considered in typography, calligraphy, and graphic design. In typography, the symmetry of letters plays a vital role in the overall aesthetics of a typeface. It involves ensuring that the left and right halves of a letter or character are evenly balanced, creating a pleasing and harmonious appearance. Letter symmetry is especially important in fonts and typefaces where each character is meticulously crafted to maintain visual consistency. It adds to the readability and visual appeal of written and printed materials.

Face Symmetry

Like, letters and shapes human face is also symmetrical in nature. If we draw a line of symmetry passing exactly through the middle of the face then the each side will appear to be same and combining them would result in the complete face. For better understanding of face symmetry the image has been added below:

Examples on Symmetry

Examples on symmetry are,

Example 1. Can the below figure considered to be symmetrical?

Solution:

No, we cannot consider this figure to be symmetrical. There is no line of symmetry existing for the figure.

Example 2: Can the below figure considered to be symmetrical?

Solution:

Yes, we can consider the figure to be symmetrical as it has a line of symmetry passing through it.

Example 3: Draw the line of symmetry of below figures if the line of symmetrical exists for below figures.

Solution:

Since, the first figure is unsymmetrical, hence the line of symmetry can be drawn for figure B only which is shown as below

Example 4: Can the below figure considered to be symmetrical?

Solution:

No this figure cannot be considered to be symmetrical in nature.

Example 5: Can the below considered to be symmetrical

Symmetry Example 5

Solution:

Yes, we can consider ‘figure A’ to be symmetrical.

Practice Questions on Symmetry

Various questions on symmetry are,

Question 1: Is the below figure symmetrical in nature?

Question 2: Is the below figure symmetrical in nature?

Question 3: Can we consider the below figure to be symmetrical in nature?

Question 4: Draw the line of symmetry of below figure if the line of symmetrical exists for below figures.

Question 5: Draw the line of symmetry of below figures if the line of symmetrical exists for below figures.

Frequently Asked Questions on Symmetry – FAQs

What is Symmetry with Examples?

Symmetry in math refers to a balance or similarity in shape, size, or arrangement on both sides of a dividing line or point. If we cut a square from its diagonal then it is symmetric.

What is Symmetry Lines?

Symmetry lines are imaginary lines that divide an object into two equal and mirror-image halves, ensuring balanced and identical reflection. They are essential in geometry and design for creating harmonious and visually appealing shapes.

How many Symmetry Lines does a Rectangle have?

A rectangle has two symmetry lines.

How many Lines of Symmetry does a Square have?

A square has four lines of symmetry, dividing it into equal halves both horizontally and vertically.

How many Lines of Symmetry does a Rhombus has?

A rhombus has 2 lines of symmetry.

What is Point of Symmetry?

A point of symmetry is a central point in a shape where the shape’s two sides are mirror images of each other. It’s the balance point that divides the shape into equal halves.

How many Lines of Symmetry does a Parallelogram have?

A parallelogram has 0 lines of symmetry, meaning it cannot be folded or rotated to match itself in any way.

How many Lines of Symmetry does a Circle has?

A circle has infinite lines of symmetry because it can be divided into two equal halves in countless ways, regardless of the angle.

What are 4 Types of Symmetry?

Four types of symmetry in mathematics are:

• Translation symmetry
• Rotational symmetry
• Reflection (or reflexive) symmetry
• Glide symmetry