NCERT Solutions for Class 7 Maths Chapter 12: Symmetry — Free PDF

Chapter 12 Overview: Symmetry

Symmetry is one of the most visually engaging chapters in Class 7 Mathematics. You will study two types of symmetry — line symmetry (mirror reflection) and rotational symmetry (turning around a centre). This chapter builds on the introduction to symmetry from Class 6 and goes deeper by introducing rotational symmetry.

Symmetry is found everywhere — in nature (butterflies, flowers, snowflakes), architecture (buildings, bridges), art (rangoli patterns, Islamic geometric art), and even in mathematics itself (graphs of functions). Understanding symmetry helps develop spatial reasoning, which is useful in geometry, design, and science.

The chapter has three exercises. Exercise 12.1 focuses on lines of symmetry for various shapes and letters. Exercise 12.2 introduces rotational symmetry and the concept of order of rotation. Exercise 12.3 brings both types together, helping you identify figures that have line symmetry, rotational symmetry, both, or neither. These concepts are tested frequently in exams, often through figures where students must identify lines of symmetry or determine the order of rotation.

Key Concepts and Definitions

Before solving the exercises, make sure you understand the following core ideas thoroughly.

Line of Symmetry (Mirror Line)

A line of symmetry is a line that divides a figure into two identical halves that are mirror images of each other. If you fold the figure along this line, both halves overlap exactly.

Key facts:
- A figure can have $0$, $1$, $2$, or more lines of symmetry.
- A regular polygon with $n$ sides has exactly $n$ lines of symmetry.
- A circle has infinitely many lines of symmetry (every diameter is a line of symmetry).
- A scalene triangle has $0$ lines of symmetry.
- Lines of symmetry can be vertical, horizontal, or diagonal.

Rotational Symmetry

A figure has rotational symmetry if it looks exactly the same after being rotated by some angle less than $360^\circ$ about a fixed point called the centre of rotation.

The smallest angle through which a figure can be rotated to look the same is called the angle of rotation. The formula is:

Order of Rotational Symmetry

The order of rotational symmetry is the number of times a figure coincides with itself (looks exactly the same) during one complete rotation of $360^\circ$.

- If order $= 1$, the figure has no rotational symmetry (it only matches at $360^\circ$, which is trivial).
- We say a figure has rotational symmetry only if the order is $\geq 2$.
- A square has order $4$ (matches at $90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$).
- An equilateral triangle has order $3$ (matches at $120^\circ$, $240^\circ$, $360^\circ$).

Figures with Both Types of Symmetry

Some figures have both line symmetry and rotational symmetry (e.g., a square), some have only one type (e.g., a parallelogram has rotational symmetry but no line symmetry), and some have neither (e.g., a scalene triangle).

For regular polygons, the number of lines of symmetry always equals the order of rotational symmetry. A regular $n$-sided polygon has $n$ lines and order $n$.

Exercise 12.1 — Lines of Symmetry

This exercise asks you to identify lines of symmetry for various shapes, letters, and figures.

Q1. Lines of symmetry for regular polygons

A regular hexagon has $6$ lines of symmetry — $3$ through opposite vertices and $3$ through midpoints of opposite sides.

General rule: A regular polygon with $n$ sides has $n$ lines of symmetry.

| Shape | Lines of Symmetry |
|---|---|
| Equilateral triangle | $3$ |
| Square | $4$ |
| Regular pentagon | $5$ |
| Regular hexagon | $6$ |
| Regular octagon | $8$ |

Q2. Lines of symmetry for common shapes

| Shape | Lines of Symmetry |
|---|---|
| Equilateral triangle | $3$ |
| Square | $4$ |
| Rectangle (non-square) | $2$ |
| Circle | Infinite |
| Isosceles triangle | $1$ |
| Scalene triangle | $0$ |
| Rhombus (non-square) | $2$ |
| Kite | $1$ |
| Semicircle | $1$ |
| Parallelogram (general) | $0$ |

Q3–Q4. Lines of symmetry for English letters

The letter A has $1$ line of symmetry — a vertical line through the middle. Letters F, G, and Z have no line of symmetry.

| Symmetry | Letters |
|---|---|
| Vertical line only | A, M, T, U, V, W, Y |
| Horizontal line only | B, C, D, E, K |
| Both vertical and horizontal | H, I, O, X |
| No line of symmetry | F, G, J, L, N, P, Q, R, S, Z |

Note: O and X have $2$ lines of symmetry each (both vertical and horizontal).

Q5. Rhombus and rectangle symmetry

A rhombus has $2$ lines of symmetry — both diagonals serve as lines of symmetry.

A rectangle has $2$ lines of symmetry — the lines joining the midpoints of opposite sides (not the diagonals). The diagonals of a rectangle are NOT lines of symmetry unless it is a square. This is because folding a rectangle along a diagonal does not make both halves overlap.

Q6. Symmetry of special figures

An arrow shape pointing right has $1$ line of symmetry (horizontal, through the middle). A plus sign (+) has $4$ lines of symmetry (horizontal, vertical, and two diagonal). A star with $5$ points (like a regular star) has $5$ lines of symmetry.

When checking for line symmetry, the practical test is: can you draw a line such that folding the figure along that line makes both halves coincide? If yes, that line is a line of symmetry.

Exercise 12.2 — Rotational Symmetry

This exercise introduces the concept of rotational symmetry and asks you to find the order and angle of rotation for various shapes.

Q1. Rotational symmetry of a square

A square looks the same at $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.

Order of rotational symmetry $= 4$.

Angle of rotation $= \dfrac{360^\circ}{4} = 90^\circ$.

Q2. Rotational symmetry of an equilateral triangle

An equilateral triangle looks the same at $120^\circ$, $240^\circ$, and $360^\circ$.

Order $= 3$. Angle of rotation $= \dfrac{360^\circ}{3} = 120^\circ$.

Q3. Rotational symmetry of a rectangle

A rectangle (non-square) looks the same at $180^\circ$ and $360^\circ$.

Order $= 2$. Angle of rotation $= \dfrac{360^\circ}{2} = 180^\circ$.

A rectangle is an example of a figure that has both line symmetry ($2$ lines) and rotational symmetry (order $2$).

Q4. Summary table for common shapes

| Shape | Order of Rotation | Angle of Rotation |
|---|---|---|
| Equilateral triangle | $3$ | $120^\circ$ |
| Square | $4$ | $90^\circ$ |
| Regular pentagon | $5$ | $72^\circ$ |
| Regular hexagon | $6$ | $60^\circ$ |
| Circle | Infinite | Any angle |
| Rectangle | $2$ | $180^\circ$ |
| Parallelogram | $2$ | $180^\circ$ |
| Rhombus | $2$ | $180^\circ$ |

Q5. Scalene triangle and the letter S

A scalene triangle only matches itself at $360^\circ$ (a full rotation). Its order is $1$, which means it has no rotational symmetry.

The letter S has rotational symmetry of order $2$ — it looks the same after a $180^\circ$ rotation. Interestingly, S has no line of symmetry. This makes S an example of a figure with rotational symmetry but no line symmetry.

Exercise 12.3 — Line Symmetry and Rotational Symmetry Together

This exercise brings both types of symmetry together and asks you to classify figures.

Q1–Q2. Square and parallelogram

A square has $4$ lines of symmetry AND rotational symmetry of order $4$. It is an example of a figure with both types of symmetry.

A parallelogram (non-rectangle, non-rhombus) has no line of symmetry, but it has rotational symmetry of order $2$ (it looks the same after a $180^\circ$ rotation). This is one of the most important exam facts from this chapter.

Q3. Complete summary table

| Figure | Lines of Symmetry | Order of Rotation |
|---|---|---|
| Equilateral triangle | $3$ | $3$ |
| Square | $4$ | $4$ |
| Regular pentagon | $5$ | $5$ |
| Regular hexagon | $6$ | $6$ |
| Circle | Infinite | Infinite |
| Rectangle (non-square) | $2$ | $2$ |
| Rhombus (non-square) | $2$ | $2$ |
| Parallelogram (general) | $0$ | $2$ |
| Isosceles triangle | $1$ | $1$ |
| Scalene triangle | $0$ | $1$ |
| Kite | $1$ | $1$ |

Q4. Rotational symmetry without line symmetry

Yes, a figure can have rotational symmetry but no line symmetry. A parallelogram (that is not a rectangle or rhombus) has rotational symmetry of order $2$ but no line of symmetry. The letter S and the letter Z also have rotational symmetry of order $2$ but no line symmetry. The recycling symbol is another example.

Q5. Line symmetry without rotational symmetry

Yes, a figure can have line symmetry but no rotational symmetry. An isosceles triangle (that is not equilateral) has $1$ line of symmetry but only has trivial rotational symmetry (order $1$). A kite also has $1$ line of symmetry but no rotational symmetry.

So the two types of symmetry are independent — having one does not guarantee the other.

Q6. Regular polygons — the connection

For a regular polygon with $n$ sides, the number of lines of symmetry equals $n$ and the order of rotational symmetry also equals $n$. They are always the same.

Examples: equilateral triangle ($3$ lines, order $3$), square ($4$ lines, order $4$), regular pentagon ($5$ lines, order $5$), regular hexagon ($6$ lines, order $6$).

Worked Examples — Additional Practice

These extra examples go beyond the textbook exercises to strengthen your understanding.

Example 1: Finding the angle of rotation

A figure has rotational symmetry of order $6$. What is the angle of rotation?

Solution:

The figure looks the same after every $60^\circ$ of rotation. It coincides with itself at $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and $360^\circ$.

Example 2: Regular octagon

How many lines of symmetry does a regular octagon ($8$ sides) have? What is its order of rotational symmetry?

Solution:
A regular octagon has $8$ lines of symmetry ($4$ through opposite vertices and $4$ through midpoints of opposite sides).

Order of rotational symmetry $= 8$.
Angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$.

Example 3: Quadrilateral identification

Name a quadrilateral that has: (a) exactly $2$ lines of symmetry (b) no lines of symmetry but rotational symmetry of order $2$.

Solution:
(a) A rectangle (non-square) or a rhombus (non-square) — both have exactly $2$ lines of symmetry.
(b) A parallelogram (that is neither a rectangle nor a rhombus) — it has no line of symmetry but has rotational symmetry of order $2$.

Example 4: Windmill blades

A windmill has $4$ blades equally spaced. What is the order of rotational symmetry? Does it have line symmetry?

Solution:
Order of rotational symmetry $= 4$ (it looks the same after $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ rotations).

If the blades are symmetric in shape, it has $4$ lines of symmetry. If each blade is curved (like a real windmill), it may have rotational symmetry but no line symmetry.

Example 5: The digit 8

The digit $8$ has how many lines of symmetry? What is its order of rotational symmetry?

Solution:
The digit $8$ has $2$ lines of symmetry — one vertical and one horizontal.
It has rotational symmetry of order $2$ (looks the same after $180^\circ$ rotation).

Common Mistakes to Avoid

Watch out for these frequent errors that cost marks in exams.

Mistake 1: Confusing diagonals with lines of symmetry

The diagonals of a rectangle are NOT lines of symmetry (unless it is a square). A line of symmetry must divide the figure into two mirror-image halves. The diagonals of a rectangle create two triangles that are congruent but are NOT mirror images across the diagonal.

Mistake 2: Saying a parallelogram has line symmetry

A general parallelogram has NO line of symmetry. This is a very commonly tested fact. Only special parallelograms (rectangles, rhombi, squares) have line symmetry. A parallelogram does have rotational symmetry of order $2$.

Mistake 3: Confusing order with angle of rotation

Order $=$ how many times the figure matches itself during a full turn. Angle $= \dfrac{360^\circ}{\text{order}}$. If order $= 4$, angle $= 90^\circ$ (not $4^\circ$). These are inversely related — higher order means smaller angle.

Mistake 4: Forgetting that order 1 means no rotational symmetry

Every figure matches itself after a $360^\circ$ rotation, so every figure has order at least $1$. We say a figure has rotational symmetry only when the order is $2$ or more. A scalene triangle has order $1$ — this does not count as rotational symmetry.

Mistake 5: Miscounting lines for the letter O

The letter O (as a perfect oval shape) has $2$ lines of symmetry (vertical and horizontal). If treated as a circle, it has infinitely many. The answer depends on how the letter is drawn — check the context of the question.

Practice Questions with Answers

Try these on your own before checking the answers below.

Q1. Regular polygon angle

A regular polygon has rotational symmetry of order $10$. How many sides does it have? What is its angle of rotation?

Answer: It has $10$ sides (regular decagon). Angle of rotation $= \dfrac{360^\circ}{10} = 36^\circ$.

Q2. Classify the shapes

For each shape, state whether it has (i) line symmetry only, (ii) rotational symmetry only, (iii) both, or (iv) neither: (a) Isosceles triangle (b) Parallelogram (c) Square (d) Scalene triangle

Answer:
(a) Line symmetry only ($1$ line, order $1$)
(b) Rotational symmetry only ($0$ lines, order $2$)
(c) Both ($4$ lines, order $4$)
(d) Neither ($0$ lines, order $1$)

Q3. Letters with rotational symmetry

Which English capital letters have rotational symmetry of order $2$?

Answer: H, I, N, O, S, X, Z all have rotational symmetry of order $2$ — they look the same after a $180^\circ$ rotation.

Q4. Designing a rangoli pattern

A rangoli pattern has $8$ lines of symmetry and rotational symmetry of order $8$. What is its angle of rotation? Name a regular polygon with the same symmetry properties.

Answer: Angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$. A regular octagon has $8$ lines of symmetry and order $8$ rotational symmetry.

Q5. True or False

(a) Every square is a rhombus, so a square has at least $2$ lines of symmetry. — True (a square has $4$ lines).
(b) A circle has exactly $360$ lines of symmetry. — False (a circle has infinitely many).
(c) The letter N has one line of symmetry. — False (N has no line of symmetry, but it has rotational symmetry of order $2$).
(d) A regular hexagon has $3$ lines of symmetry. — False (it has $6$ lines of symmetry).

Key Concepts to Remember

Tips for Scoring Full Marks

Ten Common Mistakes Students Make in Symmetry

Symmetry looks simple but it is where students lose surprise marks for carelessness. Here are ten classic errors to burn into memory.

Mistake 1: Thinking a parallelogram has line symmetry
❌ Wrong: A parallelogram has $2$ lines of symmetry.
✅ Correct: A general parallelogram has no lines of symmetry. It only has rotational symmetry of order $2$.

Mistake 2: Treating rectangle diagonals as lines of symmetry
❌ Wrong: A rectangle has $4$ lines of symmetry including its diagonals.
✅ Correct: A rectangle has $2$ lines of symmetry, running through the midpoints of opposite sides. Diagonals are not lines of symmetry except in a square.

Mistake 3: Confusing order with angle of rotation
❌ Wrong: Order of rotation $= 90$ and angle $= 4$ for a square.
✅ Correct: Order $= 4$, angle $= 90^{\circ}$. They are inversely related.

Mistake 4: Saying a scalene triangle has rotational symmetry
❌ Wrong: A scalene triangle has order $1$ rotational symmetry.
✅ Correct: Technically its order is $1$ because it matches at $360^{\circ}$, but we say it has no rotational symmetry because the order must be at least $2$ to count.

Mistake 5: Forgetting that regular polygons pair their symmetries
❌ Wrong: A regular pentagon has $5$ lines of symmetry but order $10$ rotation.
✅ Correct: For any regular $n$ sided polygon, lines of symmetry $= n$ and order of rotation $= n$.

Mistake 6: Miscounting symmetry of letters
❌ Wrong: The letter S has a vertical line of symmetry.
✅ Correct: The letter S has no line of symmetry but has order $2$ rotational symmetry.

Mistake 7: Saying a circle has $360$ lines of symmetry
❌ Wrong: A circle has $360$ lines of symmetry.
✅ Correct: A circle has infinitely many lines of symmetry. Any diameter is a line of symmetry.

Mistake 8: Saying an equilateral triangle has only $1$ line of symmetry
❌ Wrong: An equilateral triangle has $1$ line of symmetry.
✅ Correct: It has $3$ lines, one from each vertex to the midpoint of the opposite side.

Mistake 9: Wrong angle of rotation for order $6$
❌ Wrong: A figure with order $6$ has angle $6^{\circ}$.
✅ Correct: Angle $= \tfrac{360^{\circ}}{6} = 60^{\circ}$.

Mistake 10: Claiming the letter Z has line symmetry
❌ Wrong: The letter Z has a line of symmetry going through the middle.
✅ Correct: The letter Z has no lines of symmetry. It has rotational symmetry of order $2$.

Previous Year Questions (PYQs) with Solutions

These are CBSE Class 7 style questions on Symmetry from the past few years.

Q1. How many lines of symmetry does a square have? `(CBSE 2021, 1 mark)`
Solution: A square has $4$ lines of symmetry: $2$ through midpoints of opposite sides and $2$ through opposite vertices (diagonals).
Answer: $4$

Q2. State the order of rotational symmetry of an equilateral triangle. `(CBSE 2021, 1 mark)`
Solution: An equilateral triangle matches itself at $120^{\circ}, 240^{\circ}, 360^{\circ}$. Order $= 3$.
Answer: $3$

Q3. A figure has rotational symmetry of order $5$. Find the angle of rotation. `(CBSE 2022, 2 marks)`
Solution: Angle $= \tfrac{360^{\circ}}{5} = 72^{\circ}$.
Answer: $72^{\circ}$

Q4. Name a quadrilateral that has no line symmetry but has rotational symmetry of order $2$. `(CBSE 2023, 2 marks)`
Solution: A parallelogram that is neither a rectangle nor a rhombus.
Answer: Parallelogram

Q5. How many lines of symmetry does a regular hexagon have? `(CBSE 2022, 2 marks)`
Solution: A regular hexagon has $6$ lines of symmetry: $3$ through pairs of opposite vertices and $3$ through midpoints of opposite sides.
Answer: $6$

Q6. Which English capital letters have rotational symmetry of order $2$? `(CBSE 2023, 3 marks)`
Solution: H, I, N, O, S, X, Z all look the same after a $180^{\circ}$ rotation.
Answer: H, I, N, O, S, X, Z.

Q7. Does a scalene triangle have any symmetry? `(CBSE 2021, 2 marks)`
Solution: A scalene triangle has no line symmetry and no rotational symmetry (order $1$ which is trivial).
Answer: No.

Q8 (HOTS). Draw a figure which has exactly $2$ lines of symmetry and rotational symmetry of order $2$. Name at least two such figures. `(CBSE 2024, 4 marks)`
Solution: A rectangle and a rhombus both have exactly $2$ lines of symmetry and rotational symmetry of order $2$.
Answer: Rectangle, rhombus (or the digit $8$).

Practice This Chapter on SparkEd

Symmetry is one of the most visual chapters in Class 7, and the best way to get good at it is to see a lot of figures and practise spotting their symmetries. The SparkEd Symmetry module is designed to do exactly that.

Inside you will find more than 60 practice questions across three difficulty levels. Level 1 is pure recognition: count the lines of symmetry in common shapes, letters, and digits. Level 2 adds rotational symmetry, asking you to find the order and angle of rotation for polygons and real world figures. Level 3 is where you tackle CBSE style questions involving combined symmetry types, tricky quadrilaterals, and HOTS problems that ask you to design figures with specific symmetries.

The instant AI feedback is particularly useful here because symmetry mistakes are usually one off errors, like miscounting lines or confusing order with angle. The AI shows you exactly where your reasoning slipped. The gamification keeps you coming back and the adaptive engine targets the specific shapes you struggle with.

Practice Symmetry on SparkEd →

Download Symmetry worksheet

Spend fifteen minutes a day for a week and your eye will spot any line or order of symmetry at a glance.