Lines of Symmetry for Regular Polygons: An object is folded or cut into two halves, and if both the halves are mirror images of each other, then the object is said to be symmetrical. When one half of an object is the mirror image of the other, it is said to have a line of symmetry. A mirror line helps in visualising a line of symmetry.

Each regular polygon, such as an equilateral triangle or a square, has the same number of lines of symmetry as its number of sides. In this article, we will discuss the lines of symmetry for regular polygons in detail.

Introduction to Symmetry

Symmetry is a fundamental geometrical concept that may be found in nature and applied in nearly every activity. The concept of symmetry is used by artists, professionals, clothes and jewellery designers, automobile manufacturers, architects, and many others. There are symmetrical designs everywhere: beehives, flowers, tree leaves, religious symbols, apple, butterfly, carpets, and handkerchiefs.

If there is a line along which the figure may be folded so that the two portions of the figure coincide, the figure has line symmetry. We might want to keep these thoughts in mind. There are a few activities that might assist us.

Let us now enhance our symmetry concepts even further. Examine the following diagrams, which include vertical and horizontal lines to indicate symmetry lines.

The concept of line symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half. A mirror line, thus, helps to visualise a line of symmetry.

While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation.

Line of Symmetry Definition

The line of symmetry is an imaginary line or axis along which you may fold a figure to create symmetrical halves. It’s also known as the symmetry axis. Because it shows two identical reflections of an image, the line symmetry is also known as a mirror line. As a result, it’s a form of reflection symmetry. It essentially splits a thing in half. There might be one or many symmetry lines. In reality, a shape might have the following characteristics:

1. No line of symmetry implies that the figure is asymmetrical
2. One line of symmetry
3. Two lines of symmetry
4. Multiple (more than two) lines of symmetry
5. Infinite lines of symmetry

Many objects are irregular and cannot be split into equal pieces. Asymmetrical forms are those that are not symmetrical. As a result, line symmetry isn’t relevant in these situations. Let’s look at forms with different types of symmetry lines.

? Learn About Rotational Symmetry

Types of Lines of Symmetry

Any combination of vertical, horizontal, and diagonal lines or axes can be used. However, there are two distinct types of symmetry lines:

1. Vertical Line of Symmetry
2. Horizontal Line of Symmetry

1. Vertical Line of Symmetry

A vertical line of symmetry is the axis of the form that splits it into two identical halves vertically. In a vertical or straight standing stance, the mirror image of the opposite half of the form may be seen. \(A, H, M, O, U, V, W, T\), and \(Y\) are some of the alphabets that may be split vertically with symmetry.

2. Horizontal Line of Symmetry

The horizontal line of symmetry is the symmetry line or horizontal axis of a form that splits the shape into two identical halves. The axis crosses the form to divide it into two equal pieces. Horizontal symmetry may be seen in the English alphabets, such as \(B, C, D, H\), and \(E\).

Lines of Symmetry for Regular Polygons

A polygon is a closed shape made up of several line segments. A triangle is the smallest polygon with the fewest number of line segments. 

If all of the sides of a polygon have the same length and all angles are of the same measure, it is regular. As a result, an equilateral triangle is a regular three-sided polygon. Every regular polygon wil be having the number of lines of symmetry equal to their number of sides.

Lines of Symmetry in an Equilateral Triangle

Because each of its sides is of the same length and each angle is \(60^\circ \), an equilateral triangle is regular. Because an equilateral triangle has three sides, it has three symmetry lines.

Lines of Symmetry in a Quadrilateral

A square is a regular quadrilateral since all sides are the same length, and each angle is \(90^\circ\). Its diagonals can be seen as perpendicular bisectors of each other. A square has four symmetry lines, each of which divides it into two identical halves. A square’s symmetry lines include its diagonals and the lines connecting the midpoints of its opposing sides (bisectors).

Lines of Symmetry in a Regular Pentagon

A regular pentagon has five sides that are all equal. So, it contains five symmetry lines.

Lines of Symmetry in a Regular Hexagon

A regular hexagon is a six-sided polygon with equal length sides. A regular hexagon contains six lines of symmetry, as we can see in the figure below.

Lines of Symmetry in a Regular Heptagon

A regular heptagon is a seven-sided polygon with equal length sides. A regular heptagon contains seven lines of symmetry, as we can see in the figure below.

Lines of Symmetry in a Regular Octagon

A regular octagon is an eight-sided polygon with equal length sides. A regular octagon contains eight lines of symmetry, as we can see in the figure below.

And this pattern continues:

1. A regular polygon of nine sides has nine lines of symmetry
2. A regular polygon of ten sides has ten lines of symmetry
3. A regular polygon of “\(n\)” sides has “\(n\)” lines of symmetry

Line of Symmetry in a Circle

A line of symmetry of a circle is a line that passes through its centre. As a result, a circle has an infinite number of lines of symmetry.

Solved Examples on Lines of Symmetry for Regular Polygons

Q.1. Give three examples of shapes with no line of symmetry.
Ans: A scalene triangle, a parallelogram, and a trapezium do not have any line of symmetry.

Q.2. What other name can you give to the line of symmetry of 
(a) an isosceles triangle? 
(b) a circle?
Ans: a) An isosceles triangle has only one line of symmetry, which can also be called the median/altitude. 
b) There are infinite lines of symmetry in a circle, and each symmetry line is the circle’s diameter.

Q.3. State the number of lines of symmetry for the following figures: 
(a) An equilateral triangle 
(b) An isosceles triangle 
(c) A scalene triangle 
(d) A square 
(e) A rectangle 
(f) A rhombus 
Ans: (a) An equilateral triangle is a regular polygon with three sides.
Hence, the number of lines of symmetry \(=3\)
(b) An isosceles triangle is a triangle with two equal sides
Hence, the number of lines of symmetry \(=1\)
(c) A scalene triangle has all unequal sides
Hence, the number of lines of symmetry \(=0\)
 (d) A square is a regular polygon with four equal sides
 Hence, the number of lines of symmetry \(=4\)
(e) The symmetry lines in a rectangle are through the midpoints of the rectangle’s opposite and parallel lines (i.e. the bisector).
Hence, the number of lines of symmetry \(=2\)
(f) The symmetry lines of a rhombus are its diagonals. In a rhombus, the number of symmetry lines is two, i.e. the diagonals that divide it into two identical halves, each mirror image of the other.
Hence, the number of lines of symmetry \(=2\)

Q.4. What letters of the English alphabet have reflectional symmetry (i.e., symmetry-related mirror reflection). 
(a) a vertical mirror 
(b) a horizontal mirror 
(c) both horizontal and vertical mirrors
Ans: Vertical mirror – \(A, H, I, M, O, T, U, V, W, X\) and \(Y\)
(b) Horizontal mirror – \(B, C, D, E, H, I, O\) and \(X\)
(c) Both horizontal and vertical mirror – \(H, I, O\) and \(X\)

Q.5. In the following figures, the mirror line (i.e., the line of symmetry) is given as a dotted line. Complete each figure performing reflection in the dotted (mirror) line. Also, write the name of the figure you complete?

Ans: We draw the mirror image on the other side of the dotted line as shown below:

(a) This shape is a square.
(b) This shape is a triangle.
(c) This shape is a rhombus.
(d) This shape is a circle.
(e) This shape is a pentagon.
(f) This shape is an octagon

Summary

In this article, we have seen the definition of symmetry lines, types of symmetry lines, lines of symmetry in regular polygons like a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, etc. Also, we discussed the lines of symmetry in a circle.

Learn About Reflection Symmetry

FAQs

Q.1. Do all regular polygons have a line of symmetry?
Ans: Each regular polygon has a line of symmetry. The number of lines of symmetry will be equal to the number of sides of the regular polygon.

Q.2. How many lines of symmetry does a 12 sided polygon have?
Ans: A \(12\) -sided polygon has \(12\) lines of symmetry.

Q.3. What is the other name of the line of symmetry?
Ans: Another name for the line of symmetry is mirror reflection or mirror symmetry.

Q.4. Explain symmetry with an example?
Ans: Symmetry refers to something that is the same on both sides of an axis. A circle that is the same on both sides when folded along its diameter is an example of symmetry.

Q.5. Is the letter Z symmetrical?
Ans: No, the letter \(Z\) is not symmetrical.

We hope this detailed article on the lines of symmetry for regular polygons helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!