Understand what symmetry means and identify symmetrical figures
Find and draw lines of symmetry (mirror lines) in shapes, letters, and objects
Determine the number of lines of symmetry in regular polygons
Understand reflective (mirror) symmetry and complete half-drawn figures
Understand rotational symmetry and identify it in shapes and patterns
Find the order of rotational symmetry for common shapes
Recognise symmetry in Indian art — Rangoli, Mehndi, Kolam, and architecture
Create symmetrical patterns and designs
Introduction — Symmetry All Around Us
Look around you! Nature is full of beautiful, balanced shapes. A butterfly's wings are identical on both sides. A sunflower has petals arranged in a perfect circle. The Taj Mahal looks the same from the left and the right. A snowflake has six identical arms. This beautiful balance is called symmetry.
In India, symmetry is everywhere — in the Rangoli patterns drawn during Diwali, the Mehndi designs on hands, the Kolam patterns drawn in South India every morning, the Indian flag (the Ashoka Chakra has 24 identical spokes!), and the stunning architecture of Mughal monuments like the Taj Mahal and Humayun's Tomb.
In Class 5, we will learn two types of symmetry: Reflective Symmetry (mirror symmetry) and Rotational Symmetry (turning symmetry). We will also learn how to find lines of symmetry and the order of rotation for different shapes.
What is Symmetry?
A figure has symmetry if one part of the figure is an exact mirror image of the other part. When you fold a symmetrical figure along a special line, the two halves match perfectly — they overlap completely.
Definition
Symmetry: A figure is symmetrical if it can be divided into two or more identical parts that are arranged in an organised pattern.
Line of Symmetry: The line along which you can fold a figure so that the two halves match exactly. Also called a mirror line, axis of symmetry, or fold line.
The Fold Test
The simplest way to check if a figure is symmetrical is the fold test:
Draw or cut out the shape on paper.
Try to fold it along a line so that one half lies exactly on top of the other half.
If both halves match perfectly (every edge and corner overlaps), the fold line is a line of symmetry.
If the halves do NOT match, that line is NOT a line of symmetry. Try another line!
Image needed: A butterfly with a dashed vertical line down the middle showing it as a line of symmetry. Both wings are identical mirror images. Below it, show a leaf being folded along its midrib with both halves matching perfectly. Style: Clean flat illustration, child-friendly, colourful butterfly and green leaf Size: 700×400px
Example: Think of the letter A. Draw a vertical line down the middle. The left half is a mirror image of the right half. So A has a vertical line of symmetry!
Now think of the letter F. Can you fold it so both halves match? No! Whether you try a vertical, horizontal, or diagonal fold — the halves never match. So F has no line of symmetry.
Think About It: Your face is approximately symmetrical! If you draw a vertical line down the centre of your face, the left side is roughly a mirror image of the right side. But is it perfectly symmetrical? Take a photo and check — you will find tiny differences!
Line of Symmetry — Reflective Symmetry
Reflective symmetry (also called mirror symmetry or bilateral symmetry) means that one half of a figure is the mirror reflection of the other half. The line that divides the figure into two mirror-image halves is called the line of symmetry.
Lines of Symmetry Can Be:
Vertical — goes up and down (↕)
Horizontal — goes left and right (↔)
Diagonal — goes at an angle (⤡)
A figure may have one, more than one, or no lines of symmetry.
Lines of Symmetry in Common Shapes
Shape
Number of Lines of Symmetry
Where Are They?
Equilateral Triangle
3
One from each vertex to the midpoint of the opposite side
Isosceles Triangle
1
Vertical line from the top vertex to the base midpoint
Scalene Triangle
0
No sides are equal, so no lines of symmetry
Square
4
2 diagonal + 1 vertical + 1 horizontal
Rectangle
2
1 vertical + 1 horizontal (diagonals are NOT lines of symmetry!)
Rhombus
2
Along the two diagonals
Parallelogram
0
No line of symmetry (surprising but true!)
Regular Pentagon
5
One from each vertex to the midpoint of the opposite side
Regular Hexagon
6
3 through opposite vertices + 3 through midpoints of opposite sides
Circle
Infinite (∞)
Any line through the centre (diameter) is a line of symmetry
Image needed: A collection of shapes (equilateral triangle, square, rectangle, regular pentagon, regular hexagon, circle) each shown with their lines of symmetry drawn as dashed lines. Number of lines written below each shape. Colour-coded — each shape in a different colour. Style: Clean geometric illustrations, dashed lines for symmetry axes, labels below each shape Size: 800×500px
Example 1: How many lines of symmetry does a square have?
A square has 4 lines of symmetry:
• 1 vertical line (top to bottom through the centre)
• 1 horizontal line (left to right through the centre)
• 2 diagonal lines (corner to corner)
All four lines divide the square into two identical halves.
Example 2: Why does a rectangle have only 2 lines of symmetry and not 4?
A rectangle has a vertical and horizontal line of symmetry. But the diagonals are NOT lines of symmetry for a rectangle! If you fold a rectangle along its diagonal, the two halves do NOT match (one triangle hangs over the other). This is because the length and width are different.
Think About It: A parallelogram has NO lines of symmetry! Even though it looks "balanced," if you try folding it along any line — vertical, horizontal, or diagonal — the two halves never match perfectly. Surprising, isn't it?
Lines of Symmetry in English Letters and Numbers
Capital (uppercase) English letters also have symmetry! Let us group them:
Letters with a Vertical Line of Symmetry
These letters look the same if you fold them along a vertical (up-down) line:
A M T U V W Y
Letters with a Horizontal Line of Symmetry
These letters look the same if you fold them along a horizontal (left-right) line:
B C D E K
Letters with Both Vertical and Horizontal Lines of Symmetry
These special letters have 2 lines of symmetry:
H I O X
Letters with No Line of Symmetry
These letters are asymmetrical — they have no mirror line:
F G J L N P Q R S Z
Image needed: A chart showing capital letters grouped by symmetry type. Letters A, M, T, U, V, W, Y shown with vertical dashed line. Letters B, C, D, E, K shown with horizontal dashed line. Letters H, I, O, X shown with both lines. Letters F, G, J, L, N, P, Q, R, S, Z shown with a red "✗" (no symmetry). Style: Clean typography poster style, colourful groups, dashed symmetry lines clearly visible Size: 800×500px
Symmetry in Numbers (0–9)
Number
Vertical Symmetry
Horizontal Symmetry
Total Lines
0
Yes ✓
Yes ✓
2
1
Yes ✓
No ✗
1
2
No ✗
No ✗
0
3
No ✗
Yes ✓
1
4
No ✗
No ✗
0
5
No ✗
No ✗
0
6
No ✗
No ✗
0
7
No ✗
No ✗
0
8
Yes ✓
Yes ✓
2
9
No ✗
No ✗
0
Think About It: The numbers 0 and 8 are the most symmetrical digits — they have both vertical and horizontal symmetry! Can you think of a multi-digit number where the entire number is symmetrical? (Hint: try 1881 or 3003!)
Lines of Symmetry in Regular Polygons
A regular polygon is a shape where all sides are equal and all angles are equal. There is a simple rule for finding the number of lines of symmetry in regular polygons:
Golden Rule
A regular polygon with n sides has exactly n lines of symmetry.
This means: equilateral triangle (3 sides) → 3 lines; square (4 sides) → 4 lines; regular pentagon (5 sides) → 5 lines; and so on!
Regular Polygon
Number of Sides (n)
Number of Lines of Symmetry
Equilateral Triangle
3
3
Square
4
4
Regular Pentagon
5
5
Regular Hexagon
6
6
Regular Heptagon
7
7
Regular Octagon
8
8
Regular Decagon
10
10
Circle
∞ (infinite sides)
∞ (infinite lines)
Example: A regular hexagon (6 sides) has 6 lines of symmetry. Three of these go from one vertex (corner) to the opposite vertex. The other three go from the midpoint of one side to the midpoint of the opposite side.
Real-life example: A honeycomb cell is a regular hexagon! Each cell has 6 lines of symmetry, which is why honeycombs look so perfectly balanced and beautiful.
Think About It: As the number of sides in a regular polygon increases, it starts to look more and more like a circle. A regular polygon with 100 sides would look almost circular! And a circle, which you can think of as a polygon with infinite sides, has infinite lines of symmetry.
Rotational Symmetry
So far we have talked about reflective symmetry (mirror/fold symmetry). Now let us learn about a different kind of symmetry — rotational symmetry!
Definition
Rotational Symmetry: A figure has rotational symmetry if it looks exactly the same after being rotated (turned) by less than 360° (a full turn) about its centre.
Centre of Rotation: The fixed point around which the figure rotates.
Angle of Rotation: The minimum angle by which a figure must be turned to look exactly the same as the original.
Imagine pinning a shape to a board through its centre with a thumbtack. If you rotate (turn) it and it looks the same before completing a full 360° turn, it has rotational symmetry!
Example: Take a square. Pin it at the centre. Rotate it:
• After 90° (quarter turn) — it looks the same! ✓
• After 180° (half turn) — it looks the same! ✓
• After 270° (three-quarter turn) — it looks the same! ✓
• After 360° (full turn) — it looks the same! ✓
The square matches itself 4 times during one complete rotation. So the square has rotational symmetry of order 4.
Image needed: A square being rotated in 4 steps (0°, 90°, 180°, 270°) around its centre point (shown as a dot). Each position shown with a curved arrow indicating the rotation. One corner is marked with a small dot to show the rotation happening. Text labels "90°", "180°", "270°", "360°" at each step. Style: Clean diagram, 4 squares in a row showing progressive rotation, colourful arrows Size: 800×300px
Order of Rotational Symmetry
The order of rotational symmetry is the number of times a figure looks exactly the same during one complete rotation (360°).
How to Find the Order of Rotation
Order of rotation = Number of times the shape matches itself in one full turn (360°)
Angle of rotation = 360° ÷ Order
Every shape has at least order 1 (it matches itself after a full 360° turn). But we say a shape has rotational symmetry only if its order is 2 or more.
Order of Rotation for Common Shapes
Shape
Order of Rotation
Angle of Rotation
Explanation
Equilateral Triangle
3
120°
Matches after 120°, 240°, 360°
Square
4
90°
Matches after 90°, 180°, 270°, 360°
Rectangle (non-square)
2
180°
Matches after 180° and 360° only
Rhombus (non-square)
2
180°
Matches after 180° and 360° only
Parallelogram
2
180°
Matches after 180° and 360°
Regular Pentagon
5
72°
Matches after 72°, 144°, 216°, 288°, 360°
Regular Hexagon
6
60°
Matches after 60°, 120°, 180°, 240°, 300°, 360°
Regular Octagon
8
45°
Matches 8 times at equal intervals
Circle
Infinite (∞)
Any angle
Always looks the same no matter how much you rotate it
Scalene Triangle
1 (no rotational symmetry)
360°
Only matches after a full turn
Example 1: What is the order of rotation of a regular pentagon?
A regular pentagon has 5 equal sides. It matches itself 5 times in one full rotation.
Order = 5. Angle of rotation = 360° ÷ 5 = 72°.
Example 2: Does a rectangle have rotational symmetry?
Yes! A rectangle matches itself after 180° (half turn) and 360° (full turn).
Order = 2. Angle of rotation = 360° ÷ 2 = 180°.
Example 3: Does a parallelogram have rotational symmetry?
Yes! Even though a parallelogram has NO line of symmetry, it DOES have rotational symmetry of order 2! It matches itself after a 180° rotation. This is a great example of how reflective symmetry and rotational symmetry are different!
Think About It: A parallelogram has rotational symmetry (order 2) but NO reflective symmetry (0 lines of symmetry). This shows that the two types of symmetry are independent — having one does not guarantee the other!
Similarly, an isosceles triangle has reflective symmetry (1 line) but NO rotational symmetry (order 1 only). Interesting, right?
Regular Polygons — Both Types of Symmetry
Regular polygons are special because they have both reflective symmetry and rotational symmetry. Here is a summary:
Rule for Regular Polygons
A regular polygon with n sides has n lines of symmetry AND rotational symmetry of order n.
Angle of rotation = 360° ÷ n
Regular Polygon
Sides (n)
Lines of Symmetry
Order of Rotation
Angle of Rotation
Equilateral Triangle
3
3
3
120°
Square
4
4
4
90°
Regular Pentagon
5
5
5
72°
Regular Hexagon
6
6
6
60°
Regular Octagon
8
8
8
45°
Regular Decagon
10
10
10
36°
Example: The Ashoka Chakra on the Indian flag has 24 spokes equally spaced. It behaves like a regular 24-gon! So it has:
• Lines of symmetry = 24
• Order of rotation = 24
• Angle of rotation = 360° ÷ 24 = 15°
This is why the Ashoka Chakra looks the same even after a small rotation — it has very high symmetry!
Symmetry in Indian Art and Architecture
India has a rich tradition of art and architecture that uses symmetry beautifully. Let us explore some examples:
1. Rangoli Patterns
Rangoli is a traditional Indian art form drawn on the floor during festivals like Diwali, Pongal, and Onam. Most Rangoli designs have multiple lines of symmetry (often 4, 6, or 8) and high rotational symmetry. A typical Rangoli has both vertical, horizontal, and diagonal lines of symmetry.
2. Kolam Patterns (South India)
Kolam is drawn every morning in front of homes in Tamil Nadu, Karnataka, and Andhra Pradesh. Kolam designs are drawn around a grid of dots and usually have 4 lines of symmetry (vertical, horizontal, and two diagonals) with rotational symmetry of order 4.
3. Mehndi (Henna) Designs
Mehndi designs drawn on hands for weddings and festivals often use reflective symmetry. The design on the palm usually has a vertical line of symmetry, with the pattern on the left mirroring the pattern on the right.
4. Mughal Architecture
The Taj Mahal in Agra is one of the most perfectly symmetrical buildings in the world. It has a vertical line of symmetry — the left half is a mirror image of the right half. The gardens (Charbagh style) have 4 lines of symmetry — they are divided into four equal quadrants.
5. Warli Art (Maharashtra)
Warli tribal art uses simple geometric shapes — circles, triangles, and squares — arranged in symmetrical patterns. Many Warli motifs have rotational symmetry, especially the circular dance formations.
Image needed: A collage showing 4 examples of symmetry in Indian art: (1) A colourful Rangoli with 4 lines of symmetry marked, (2) A Kolam dot pattern with symmetry lines, (3) The Taj Mahal with a vertical line of symmetry marked, (4) A Mehndi design on a hand. All with dashed lines showing symmetry axes. Style: Colourful flat illustrations, traditional Indian art motifs, dashed symmetry lines Size: 800×500px
Real-Life Challenge: Next time you see a Rangoli or Kolam, count its lines of symmetry! A simple four-petal flower Rangoli has 4 lines of symmetry. A six-pointed star Rangoli (hexagonal) has 6 lines of symmetry. Can you find a Rangoli with 8 lines of symmetry?
Completing Symmetrical Figures
A common exercise in symmetry is to complete a half-drawn figure given its line of symmetry. Here is how to do it:
Steps to Complete a Symmetrical Figure
Step 1: Identify the line of symmetry (mirror line).
Step 2: For each point on the given half, measure its distance from the mirror line.
Step 3: Plot the mirror image point at the same distance on the other side of the line.
Step 4: Connect the mirror points in the same order to complete the figure.
Example: If a point is 3 cm to the left of the mirror line, its reflection will be 3 cm to the right of the mirror line (at the same height).
Using a grid: On a square grid (graph paper), if a point is 4 squares to the left of the vertical mirror line, draw its reflection 4 squares to the right, on the same horizontal row.
Think About It: Mirror writing is a fun application of reflective symmetry! If you write a word and hold it up to a mirror, the mirror shows its reflection. Leonardo da Vinci famously wrote all his notes in mirror writing! In India, the word "AMBULANCE" is written in mirror writing on ambulances so that drivers see it correctly in their rear-view mirrors.
Key Words
Term
Meaning
Symmetry
A balanced arrangement where one part is a mirror image or rotated copy of another part
Line of Symmetry
The line that divides a figure into two mirror-image halves (also called mirror line or axis of symmetry)
Reflective Symmetry
Symmetry where one half is the reflection (mirror image) of the other half
Rotational Symmetry
Symmetry where a figure looks the same after rotation by less than 360°
Centre of Rotation
The fixed point around which a figure rotates
Order of Rotation
The number of times a figure matches itself in one full turn (360°)
Angle of Rotation
The smallest angle by which a figure is rotated to match itself (= 360° ÷ order)
Regular Polygon
A polygon with all sides equal and all angles equal
Asymmetrical
Having no symmetry (no line of symmetry and rotational order of 1)
Mirror Image
A reflection of a figure across a line — left becomes right and vice versa
Tessellation
A pattern of shapes that fit together without gaps or overlaps, covering a surface completely
Key Points to Remember
A figure is symmetrical if one half is the exact mirror image of the other half.
The line of symmetry (mirror line) divides a figure into two identical halves that overlap when folded.
A shape can have 0, 1, 2, 3, 4, or more lines of symmetry. A circle has infinite lines.
A regular polygon with n sides has exactly n lines of symmetry.
Rotational symmetry means the shape looks the same after turning by less than 360°.
Order of rotation = number of times the shape matches itself in one full turn.
Angle of rotation = 360° ÷ order of rotation.
A regular polygon with n sides has rotational symmetry of order n.
A parallelogram has rotational symmetry (order 2) but NO line of symmetry.
A scalene triangle has neither reflective nor rotational symmetry.
The rectangle has 2 lines of symmetry — vertical and horizontal (NOT diagonal!).
Indian art forms like Rangoli, Kolam, Mehndi use both types of symmetry beautifully.
Finished learning? Now test what you know!
🎯 Quick Quiz — Test Yourself!
Answer these questions right here. You get instant feedback — no paper needed!
Score: 0 / 8
1. A line that divides a shape into two matching halves is a line of:
A line of symmetry splits a shape into two mirror-image halves.
2. How many lines of symmetry does a square have?
A square has 4 lines of symmetry (2 diagonals + 2 through mid-sides).
3. How many lines of symmetry does a circle have?
Every line through the centre of a circle is a line of symmetry — countless many.
4. Which letter has a vertical line of symmetry?
A is the same on both sides of a vertical line. F, G and P are not.
5. How many lines of symmetry does an equilateral triangle have?
An equilateral triangle has 3 lines of symmetry.
6. How many lines of symmetry does a rectangle (non-square) have?
A rectangle has 2 lines of symmetry (through the middle of each pair of opposite sides).
7. A regular hexagon has how many lines of symmetry?
A regular polygon with n sides has n lines of symmetry — a hexagon has 6.
8. A square has rotational symmetry of what order (how many times it looks the same in one full turn)?
A square looks the same 4 times in a full 360° turn — order 4.
Practice Questions
A. Write the number of lines of symmetry for each shape:
1. Equilateral triangle =
2. Square =
3. Rectangle =
4. Regular hexagon =
5. Circle =
6. Parallelogram =
7. Isosceles triangle =
8. Regular octagon =
9. Scalene triangle =
10. Rhombus =
B. Draw all lines of symmetry for these shapes:
1. Draw a square and show all 4 lines of symmetry.
2. Draw a rectangle and show both lines of symmetry. Explain why the diagonals are NOT lines of symmetry.
3. Draw a regular hexagon and show all 6 lines of symmetry.
4. Draw an equilateral triangle and show all 3 lines of symmetry.
C. Find the order of rotational symmetry:
1. Equilateral triangle — Order = , Angle =
2. Square — Order = , Angle =
3. Regular pentagon — Order = , Angle =
4. Regular hexagon — Order = , Angle =
5. Rectangle — Order = , Angle =
6. Parallelogram — Order = , Angle =
7. Scalene triangle — Order = , Angle =
8. Circle — Order = , Angle =
D. Lines of symmetry in letters and numbers:
1. Write all capital letters that have exactly ONE vertical line of symmetry.
2. Write all capital letters that have exactly ONE horizontal line of symmetry.
3. Write all capital letters that have BOTH vertical and horizontal lines of symmetry.
4. Write all capital letters that have NO line of symmetry.
5. Which digits (0–9) have a vertical line of symmetry?
6. Which digits (0–9) have a horizontal line of symmetry?
E. True or False:
1. A square has exactly 4 lines of symmetry.
2. The diagonals of a rectangle are lines of symmetry.
3. A circle has exactly 10 lines of symmetry.
4. A parallelogram has no line of symmetry.
5. A regular hexagon has rotational symmetry of order 6.
6. Every shape with a line of symmetry also has rotational symmetry.
7. A parallelogram has rotational symmetry of order 2.
8. The letter Z has no line of symmetry but has rotational symmetry.
F. Choose the correct answer (MCQ):
1. How many lines of symmetry does a regular pentagon have?
a) 3b) 4c) 5d) 10
2. The order of rotational symmetry of a rectangle is:
a) 1b) 2c) 4d) 0
3. Which shape has rotational symmetry but NO line of symmetry?
a) Squareb) Parallelogramc) Circled) Equilateral triangle
4. The angle of rotation for a regular hexagon is:
a) 90°b) 72°c) 60°d) 45°
5. How many capital English letters have both vertical and horizontal lines of symmetry?
a) 2b) 3c) 4d) 5
6. A regular polygon has 10 lines of symmetry. How many sides does it have?
a) 5b) 8c) 10d) 20
G. Fill in the blanks:
1. A regular polygon with n sides has lines of symmetry.
2. The order of rotational symmetry of a regular polygon with n sides is .
3. Angle of rotation = 360° ÷ .
4. A square has lines of symmetry and rotational symmetry of order .
5. A rectangle has lines of symmetry and rotational symmetry of order .
6. A figure that has no line of symmetry and order of rotation 1 is called .
7. The Ashoka Chakra has 24 spokes, so its order of rotational symmetry is .
8. A circle has lines of symmetry and rotational symmetry of order .
H. Word Problems:
1. Meena is making a Rangoli design for Diwali. Her Rangoli has 8 lines of symmetry and rotational symmetry of order 8. What regular polygon shape is her Rangoli based on? What is its angle of rotation?
2. Arjun says that the diagonals of a rectangle are lines of symmetry. Dhruv says they are not. Who is correct? Explain why using the fold test.
3. A fan has 5 identical blades equally spaced. What is its order of rotational symmetry? What is the angle between two consecutive blades?
4. Priya drew a shape that has exactly 1 line of symmetry and no rotational symmetry (order 1). Give two examples of such a shape.
5. The wheel of Konark Sun Temple has 8 spokes. What is its order of rotational symmetry? What is the angle of rotation?
Activity — Create a Symmetrical Rangoli!
Make your own symmetrical Rangoli using the paper-folding method:
Take a square piece of paper.
Fold it in half vertically (you now have a rectangle).
Fold it in half again horizontally (you now have a smaller square).
Fold it diagonally (you now have a triangle — the paper is folded into 8 layers).
Cut shapes along the edges — small triangles, curves, or zigzag patterns. Be creative!
Unfold the paper carefully.
See the magic! You have a beautiful symmetrical pattern with 4 (or 8) lines of symmetry!
Questions about your Rangoli:
How many lines of symmetry does your Rangoli have?
What is the order of rotational symmetry?
What is the angle of rotation?
Bonus: Colour your Rangoli symmetrically! If one section is red, the mirror image section should also be red. This ensures the colours maintain the symmetry.
Want to review the lesson again?
Mathematics
Symmetry and Rotational Patterns
Class 5 | CBSE / NCERT / ICSE
Name: Class: Date:
Learning Objectives
Understand symmetry and identify symmetrical figures
Find and draw lines of symmetry in shapes, letters, and objects
Determine lines of symmetry in regular polygons
Understand reflective (mirror) symmetry and rotational symmetry
Find the order of rotational symmetry for common shapes
Recognise symmetry in Indian art — Rangoli, Mehndi, Kolam
Introduction — Symmetry All Around Us
A butterfly's wings are identical on both sides. The Taj Mahal looks the same from left and right. Rangoli and Kolam patterns are perfectly balanced. This beautiful balance is called symmetry. In Class 5, we learn two types: Reflective Symmetry (mirror symmetry) and Rotational Symmetry (turning symmetry).
What is Symmetry?
A figure has symmetry if one part is an exact mirror image of the other. When folded along a special line, both halves match perfectly.
Line of Symmetry (Mirror Line): The line along which a figure can be folded so that both halves overlap exactly. It can be vertical, horizontal, or diagonal.
Fold Test: Cut out the shape → fold along a line → if both halves match perfectly → that line is a line of symmetry!
[Image: Butterfly with vertical line of symmetry marked; leaf being folded along midrib]
Lines of Symmetry in Common Shapes
Shape
Lines of Symmetry
Where?
Equilateral Triangle
3
From each vertex to midpoint of opposite side
Isosceles Triangle
1
Vertical: vertex to base midpoint
Scalene Triangle
0
No equal sides → no symmetry
Square
4
2 diagonal + 1 vertical + 1 horizontal
Rectangle
2
1 vertical + 1 horizontal (NOT diagonals!)
Rhombus
2
Along both diagonals
Parallelogram
0
No line of symmetry!
Regular Pentagon
5
One from each vertex
Regular Hexagon
6
3 vertex-to-vertex + 3 midpoint-to-midpoint
Circle
∞ (infinite)
Every diameter is a line of symmetry
[Image: Shapes (triangle, square, rectangle, pentagon, hexagon, circle) with lines of symmetry drawn as dashed lines, number labelled below each]
Golden Rule: A regular polygon with n sides has exactly n lines of symmetry.
Lines of Symmetry in Letters and Numbers
Symmetry Type
Letters
Vertical line only
A, M, T, U, V, W, Y
Horizontal line only
B, C, D, E, K
Both vertical & horizontal
H, I, O, X
No line of symmetry
F, G, J, L, N, P, Q, R, S, Z
Numbers with symmetry: 0, 8 (both lines); 1 (vertical); 3 (horizontal). Numbers 2, 4, 5, 6, 7, 9 have no line of symmetry.
[Image: Capital letters grouped by symmetry type with dashed lines showing symmetry axes]
Rotational Symmetry
Rotational Symmetry: A figure has rotational symmetry if it looks exactly the same after being rotated (turned) by less than 360° about its centre. Order of Rotation: Number of times the shape matches itself in one full turn (360°). Angle of Rotation: 360° ÷ Order of rotation.
Example: A square rotated about its centre matches itself at 90°, 180°, 270°, and 360°. So it has rotational symmetry of order 4 and angle of rotation = 360° ÷ 4 = 90°.
[Image: A square being rotated in 4 steps (0°, 90°, 180°, 270°) around centre point, with one corner marked to show rotation]
Order of Rotation — Common Shapes
Shape
Order
Angle
Note
Equilateral Triangle
3
120°
Matches at 120°, 240°, 360°
Square
4
90°
Matches at 90°, 180°, 270°, 360°
Rectangle
2
180°
Matches at 180° and 360°
Rhombus
2
180°
Matches at 180° and 360°
Parallelogram
2
180°
Has rotation but NO reflection symmetry!
Regular Pentagon
5
72°
360° ÷ 5 = 72°
Regular Hexagon
6
60°
360° ÷ 6 = 60°
Regular Octagon
8
45°
360° ÷ 8 = 45°
Circle
∞
Any
Always matches
Scalene Triangle
1
360°
No rotational symmetry
Important: A parallelogram has rotational symmetry (order 2) but NO line of symmetry. An isosceles triangle has 1 line of symmetry but NO rotational symmetry (order 1). The two types are independent!
Regular Polygons — Both Types of Symmetry
Rule: A regular polygon with n sides → n lines of symmetry + rotational symmetry of order n + angle = 360° ÷ n.
Polygon
Sides
Lines of Sym.
Order
Angle
Equil. Triangle
3
3
3
120°
Square
4
4
4
90°
Reg. Pentagon
5
5
5
72°
Reg. Hexagon
6
6
6
60°
Reg. Octagon
8
8
8
45°
Example: The Ashoka Chakra on the Indian flag has 24 spokes → Order = 24, Angle = 360° ÷ 24 = 15°. The Konark Sun Temple wheel has 8 spokes → Order = 8, Angle = 45°.
Symmetry in Indian Art
Rangoli: Multiple lines of symmetry (4, 6, or 8) + high rotational symmetry
Kolam: Usually 4 lines of symmetry with rotational order 4
Mehndi: Vertical reflective symmetry on palm designs
Taj Mahal: 1 vertical line of symmetry; Charbagh garden has 4 lines
Warli Art: Circular dance motifs with rotational symmetry
[Image: Collage — Rangoli with symmetry lines, Kolam pattern, Taj Mahal with vertical line, Mehndi design]
Completing Symmetrical Figures
Steps: (1) Find the mirror line → (2) Measure distance of each point from the line → (3) Plot mirror point at same distance on other side → (4) Connect points to complete figure.
Key Words
Term
Meaning
Symmetry
Balanced arrangement — one part mirrors or rotates to match another
Line of Symmetry
Fold line dividing figure into two matching halves
Reflective Symmetry
One half is the mirror image of the other
Rotational Symmetry
Figure matches itself after rotation < 360°
Centre of Rotation
Fixed point around which figure rotates
Order of Rotation
Times figure matches itself in one full turn
Angle of Rotation
360° ÷ order
Regular Polygon
All sides equal, all angles equal
Asymmetrical
No symmetry of any kind
Tessellation
Pattern of shapes covering a surface with no gaps
Key Points to Remember:
• Line of symmetry = fold line where both halves match exactly.
• Regular polygon with n sides → n lines of symmetry + order n rotation.
• Angle of rotation = 360° ÷ order.
• Parallelogram: order 2 rotation, but 0 lines of symmetry.
• Rectangle: 2 lines (vertical + horizontal), diagonals are NOT lines of symmetry.
• Circle: infinite lines, infinite order of rotation.
• Indian art (Rangoli, Kolam, Mehndi, Mughal architecture) uses symmetry beautifully.
Practice Questions
A. Write the number of lines of symmetry:
1. Equilateral triangle =
2. Square =
3. Rectangle =
4. Regular hexagon =
5. Circle =
6. Parallelogram =
7. Isosceles triangle =
8. Regular octagon =
9. Scalene triangle =
10. Rhombus =
B. Draw all lines of symmetry for these shapes:
1. Draw a square and show all 4 lines of symmetry.
2. Draw a rectangle and show both lines. Why are diagonals NOT lines of symmetry?
3. Draw a regular hexagon and show all 6 lines of symmetry.
4. Draw an equilateral triangle and show all 3 lines of symmetry.
C. Find the order of rotational symmetry:
1. Equilateral triangle — Order = , Angle =
2. Square — Order = , Angle =
3. Regular pentagon — Order = , Angle =
4. Regular hexagon — Order = , Angle =
5. Rectangle — Order = , Angle =
6. Parallelogram — Order = , Angle =
7. Scalene triangle — Order = , Angle =
8. Circle — Order = , Angle =
D. Lines of symmetry in letters:
1. Letters with vertical symmetry only:
2. Letters with horizontal symmetry only:
3. Letters with both vertical & horizontal:
4. Letters with no symmetry:
5. Digits (0–9) with vertical symmetry:
6. Digits (0–9) with horizontal symmetry:
E. True or False:
1. A square has exactly 4 lines of symmetry.
2. The diagonals of a rectangle are lines of symmetry.
3. A circle has exactly 10 lines of symmetry.
4. A parallelogram has no line of symmetry.
5. A regular hexagon has rotational symmetry of order 6.
6. Every shape with a line of symmetry also has rotational symmetry.
7. A parallelogram has rotational symmetry of order 2.
8. The letter Z has no line of symmetry but has rotational symmetry.
F. Choose the correct answer (MCQ):
1. Lines of symmetry in a regular pentagon?
a) 3b) 4c) 5d) 10
2. Order of rotational symmetry of a rectangle?
a) 1b) 2c) 4d) 0
3. Which has rotational symmetry but NO line of symmetry?
a) Squareb) Parallelogramc) Circled) Equil. triangle
4. Angle of rotation for a regular hexagon?
a) 90°b) 72°c) 60°d) 45°
5. Capital letters with both vertical and horizontal symmetry?
a) 2b) 3c) 4d) 5
6. A regular polygon has 10 lines of symmetry. Sides?
a) 5b) 8c) 10d) 20
G. Fill in the blanks:
1. A regular polygon with n sides has lines of symmetry.
2. Order of rotational symmetry of a regular n-sided polygon = .
3. Angle of rotation = 360° ÷ .
4. A square has lines of symmetry and order .
5. A rectangle has lines of symmetry and order .
6. No symmetry at all → the figure is called .
7. Ashoka Chakra (24 spokes) → order = .
8. A circle has lines of symmetry and order .
H. Word Problems:
1. Meena's Rangoli has 8 lines of symmetry and order 8 rotation. What polygon shape is it based on? What is the angle of rotation?
2. Arjun says rectangle diagonals are lines of symmetry. Dhruv says no. Who is correct? Explain.
3. A fan has 5 identical blades equally spaced. Order of rotational symmetry? Angle between blades?
4. Priya's shape has 1 line of symmetry and no rotational symmetry. Give 2 examples.
5. Konark Sun Temple wheel has 8 spokes. Order? Angle of rotation?
Activity — Create a Symmetrical Rangoli!
Take a square paper → Fold in half vertically → Fold again horizontally → Fold diagonally → Cut shapes along edges → Unfold → See your symmetrical pattern!
Answer about your Rangoli:
Lines of symmetry: | Order of rotation: | Angle:
Bonus: Colour it symmetrically — matching sections get matching colours!
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