Symmetry is all around us. The human face has a natural symmetry, a butterfly has symmetric wings, and many buildings and monuments like the Taj Mahal are famous for their symmetry. A figure is said to be symmetric if it can be divided into two identical halves by a line. This line is called the line of symmetry or the axis of symmetry.
When we fold a symmetric figure along its line of symmetry, the two halves fall exactly on top of each other. For example, if we fold a square along one of its diagonals, the two triangular halves match exactly. Symmetry is not only beautiful; it also helps engineers design balanced structures and artists create pleasing patterns.
In this chapter, we learn to identify lines of symmetry in different figures, count the number of lines of symmetry in regular polygons, and understand reflection symmetry. We also learn how to recognise symmetric figures in our environment and how to complete figures that have symmetry.
2. Line of Symmetry
A line of symmetry is a line along which a figure can be folded such that the two halves match exactly. A figure can have:
No line of symmetry, like a scalene triangle.
One line of symmetry, like an isosceles triangle.
Two lines of symmetry, like a rectangle.
Many lines of symmetry, like a circle which has infinitely many.
To find a line of symmetry, we imagine folding the figure. If the two parts coincide perfectly, the fold line is a line of symmetry.
Examples
A scalene triangle has no line of symmetry.
An isosceles triangle has one line of symmetry.
An equilateral triangle has three lines of symmetry.
A rectangle has two lines of symmetry.
A square has four lines of symmetry.
A circle has infinitely many lines of symmetry.
3. Symmetric and Asymmetric Figures
A figure is symmetric if it has at least one line of symmetry. A figure that has no line of symmetry is called asymmetric.
For example, the letters A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y have vertical or horizontal lines of symmetry. The letter F is asymmetric because it has no line of symmetry.
Regular Polygons
A regular polygon has all sides equal and all angles equal. A regular polygon with n sides has n lines of symmetry.
Equilateral triangle (3 sides) has 3 lines of symmetry.
Square (4 sides) has 4 lines of symmetry.
Regular pentagon (5 sides) has 5 lines of symmetry.
Regular hexagon (6 sides) has 6 lines of symmetry.
4. Reflection Symmetry
Reflection symmetry (also called mirror symmetry) means that one half of a figure is the mirror image of the other half. A line of symmetry acts like a mirror: the reflection of one half gives the other half.
When you place a mirror on a line of symmetry, the image in the mirror completes the figure. In reflection:
A point on the figure and its mirror image are at equal distances from the line of symmetry.
The line joining a point to its image is perpendicular to the line of symmetry.
The distance of a point from the mirror line is equal to the distance of its image from the mirror line.
Reflection and Orientation
Reflection reverses the orientation of a figure. For example, the letter 'b' becomes 'd' when reflected, and the digits 6 and 9 reverse their orientation.
5. Lines of Symmetry in Common Figures
Triangle
Scalene triangle: no line of symmetry.
Isosceles triangle: one line of symmetry.
Equilateral triangle: three lines of symmetry.
Quadrilaterals
Rectangle: two lines of symmetry (through the midpoints of opposite sides).
Square: four lines of symmetry (two diagonals and two midlines).
Parallelogram: no lines of symmetry (unless it is a rectangle or rhombus).
Rhombus: two lines of symmetry (the diagonals).
Trapezium: an isosceles trapezium has one line of symmetry.
Circle
A circle has infinitely many lines of symmetry because every line passing through its centre divides it into two identical halves.
6. Symmetry in the Alphabet and Digits
Many capital letters have lines of symmetry:
- Vertical line of symmetry: A, H, I, M, O, T, U, V, W, X, Y.
- Horizontal line of symmetry: B, C, D, E, H, I, K, O, X.
- Letters with no line of symmetry: F, G, J, L, N, P, Q, R, S, Z.
Among digits, 0 and 8 have vertical lines of symmetry, and 3 has a horizontal line of symmetry.
7. Completing Symmetric Figures
When half of a figure and its line of symmetry are given, we can complete the figure by drawing the mirror image of the given half. To do this:
Identify the line of symmetry.
For every point of the given half, mark its mirror point at an equal distance on the other side of the line.
Join the mirror points in the same order.
8. Symmetry in Nature and Architecture
Symmetry appears in leaves, flowers, butterflies, snowflakes and the human body. In architecture, symmetric designs create balance and are used in buildings, bridges and monuments. Understanding symmetry helps us appreciate the balance in nature and design.
Quick Revision Tables
Figure
Number of Lines of Symmetry
Scalene triangle
0
Isosceles triangle
1
Equilateral triangle
3
Rectangle
2
Square
4
Regular pentagon
5
Regular hexagon
6
Circle
Infinite
Letter
Line of Symmetry
A
Vertical
B
Horizontal
H
Both vertical and horizontal
F
None
Mind Map
flowchart TD
A["Symmetry"] --> B["Line of Symmetry"]
A --> C["Symmetric and Asymmetric Figures"]
A --> D["Reflection Symmetry"]
A --> E["Regular Polygons"]
A --> F["Completing Symmetric Figures"]
B --> B1["Fold line divides into equal halves"]
C --> C1["At least one line of symmetry"]
C --> C2["Asymmetric: no line of symmetry"]
D --> D1["Mirror image on other side"]
D --> D2["Equal distance from the line"]
E --> E1["n sides give n lines of symmetry"]
F --> F1["Mark mirror points, join them"]
A --> G["Nature and architecture"]
Important Diagrams (SVG)
Lines of Symmetry in a Square
Reflection Symmetry of a Butterfly
Common Mistakes
Thinking every triangle has at least one line of symmetry. A scalene triangle has no line of symmetry.
Confusing the number of lines of symmetry of a rectangle (2) with that of a square (4).
Believing a parallelogram has lines of symmetry. A general parallelogram has none.
Forgetting that a circle has infinitely many lines of symmetry, not just one or two.
Confusing reflection symmetry with rotational symmetry.
Drawing a line of symmetry that does not divide the figure into equal halves.
Exam Tips
Fold the figure mentally (or on paper in the exam) to check whether the halves match.
Memorise the lines of symmetry for triangles, quadrilaterals and regular polygons.
Remember the rule: a regular polygon with n sides has n lines of symmetry.
For letters and digits, practise identifying vertical and horizontal lines of symmetry.
When completing a symmetric figure, mark mirror points at equal distances from the line.
In reflection, remember that distances from the mirror line are preserved.
Conclusion
Symmetry introduces us to the beautiful balance found in shapes, nature and design. We learnt to identify lines of symmetry, classify figures as symmetric or asymmetric, and understand reflection symmetry where one half is the mirror image of the other. The rule that a regular polygon with n sides has n lines of symmetry helps us count them easily. Symmetry is used in art, architecture and science, making it one of the most visually appealing topics in mathematics.