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Symmetry

NCERT Class 5 · Mathematics Based on NCERT Class 5 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores two types of symmetry: reflection symmetry (line symmetry) and rotational symmetry. Students learn how to identify lines of symmetry in letters, numbers, and shapes, and how to create symmetrical designs by folding and cutting paper. The big idea is that a shape has symmetry if it looks the same after a reflection (flip) or a rotation (turn), and understanding this helps in making patterns and cutouts.
Key Concepts (10)
Reflection symmetry
A shape has reflection symmetry if it can be divided into two identical halves that are mirror images of each other along a line called the line of symmetry.
e.g. The letter 'A' has a vertical line of symmetry. If you fold a paper along that line, both halves match exactly.
Line of symmetry
A line that divides a shape into two identical halves that are mirror images of each other.
e.g. The letter 'H' has two lines of symmetry: one vertical and one horizontal.
Vertical line of symmetry
A line of symmetry that runs from top to bottom, dividing the shape into left and right mirror halves.
e.g. The letter 'A' has a vertical line of symmetry.
Horizontal line of symmetry
A line of symmetry that runs from left to right, dividing the shape into top and bottom mirror halves.
e.g. The letter 'H' has a horizontal line of symmetry.
Rotational symmetry
A shape has rotational symmetry if it looks exactly the same after being rotated by a certain angle (less than a full turn) about a central point.
e.g. A firki (windmill) looks the same after a 1/4 turn, 1/2 turn, and 3/4 turn.
Half turn
A rotation of 180 degrees (1/2 of a full turn).
e.g. The letter 'H' looks the same after a half turn.
Quarter turn
A rotation of 90 degrees (1/4 of a full turn).
e.g. A firki looks the same after a quarter turn.
Full turn
A rotation of 360 degrees, bringing the shape back to its original position.
e.g. Every shape looks the same after a full turn.
Symmetrical design
A pattern or shape that has either reflection symmetry, rotational symmetry, or both.
e.g. Block printing designs from Rajasthan often have both reflection and rotational symmetry.
Firki
A windmill toy made by folding a square paper and attaching it to a stick, which has rotational symmetry.
e.g. The firki looks the same after 1/4, 1/2, and 3/4 turns.
Formulas & Equations (1)
No formulas in this chapter
Reads as: Not applicable
Variables: Not applicable
Use when: Not applicable
⚠ Not applicable
How It Works
Making a letter cutout using reflection symmetry (e.g., letter A)
  1. Step 1: Fold a paper in half.
  2. Step 2: Draw half of the letter A along the fold.
  3. Step 3: Cut along the outline.
  4. Step 4: Open the paper to see the full letter A.
💡 The fold line becomes the line of symmetry. Only draw half the letter; the fold creates the mirror image.
Making a letter cutout with two lines of symmetry (e.g., letter H)
  1. Step 1: Fold the paper into one-fourth (once vertically, once horizontally).
  2. Step 2: Draw one-fourth of the letter H along the folds.
  3. Step 3: Cut along the outline.
  4. Step 4: Open the paper to see the full letter H.
💡 Both folds are lines of symmetry. Drawing only one-fourth ensures the full shape has both vertical and horizontal symmetry.
Making a firki (windmill)
  1. Step 1: Take a square paper.
  2. Step 2: Fold the paper in half diagonally to make two triangles.
  3. Step 3: Open and fold it the other way to make two more triangles.
  4. Step 4: Open it again to see an 'X' shape on the paper.
  5. Step 5: Use scissors to cut along the four lines of the 'X', stopping about halfway to the centre.
  6. Step 6: Take one corner of each triangle and fold it gently towards the centre. Do not press it flat.
  7. Step 7: Fold every other corner towards the centre.
  8. Step 8: Push a pin through the folded corners and the centre of the paper.
  9. Step 9: Push the pin through a stick or straw.
💡 The firki has rotational symmetry — it looks the same after 1/4, 1/2, and 3/4 turns. Do not press the folds flat, or the firki will not spin.
Checking rotational symmetry of a shape
  1. Step 1: Mark a dot on the shape to track its orientation.
  2. Step 2: Rotate the shape by 1/4 turn and check if it looks the same.
  3. Step 3: Rotate by 1/2 turn and check.
  4. Step 4: Rotate by 3/4 turn and check.
  5. Step 5: Rotate by a full turn (always looks the same).
💡 A shape has rotational symmetry if it looks the same at any turn less than a full turn. The letter H has rotational symmetry only at 1/2 turn.
Block printing with symmetrical blocks
  1. Step 1: Carve a wooden block with a symmetrical design.
  2. Step 2: Dip the block in colour.
  3. Step 3: Press the block onto fabric to create a print.
  4. Step 4: Repeat the block to create a pattern.
💡 The block itself may have reflection or rotational symmetry. The final print is made by using the block multiple times.
Exam Traps — Don't Make These Mistakes
⚠ Confusing reflection symmetry with rotational symmetry
✗ The letter 'A' has rotational symmetry because it looks the same when turned upside down.
→ The letter 'A' does NOT look the same after any rotation (except a full turn). It only has reflection symmetry (vertical line).
✓ The letter 'A' has a vertical line of symmetry (reflection symmetry) but no rotational symmetry.
⚠ Thinking every shape with a line of symmetry also has rotational symmetry
✗ The letter 'V' has rotational symmetry because it has a vertical line of symmetry.
→ Having a line of symmetry does not guarantee rotational symmetry. The letter 'V' looks different when rotated by 1/2 turn (it becomes upside down).
✓ The letter 'V' has a vertical line of symmetry (reflection symmetry) but does not have rotational symmetry.
⚠ Mistaking a full turn for rotational symmetry
✗ Every shape has rotational symmetry because it looks the same after a full turn.
→ Rotational symmetry means the shape looks the same at a turn LESS than a full turn (e.g., 1/4, 1/2, or 3/4 turn). A full turn always returns to the original, so it does not count.
✓ A shape has rotational symmetry only if it looks the same after a 1/4, 1/2, or 3/4 turn (not just a full turn).
⚠ Incorrectly identifying lines of symmetry in letters
✗ The letter 'N' has a vertical line of symmetry.
→ The letter 'N' does NOT have any line of symmetry. Its left and right halves are not mirror images.
✓ The letter 'N' has no reflection symmetry and no rotational symmetry.
⚠ Forgetting that a shape can have both symmetries
✗ The letter 'H' has only reflection symmetry.
→ The letter 'H' has both reflection symmetry (vertical and horizontal lines) AND rotational symmetry (looks the same after a 1/2 turn).
✓ The letter 'H' has two lines of symmetry (vertical and horizontal) and rotational symmetry at a 1/2 turn.
Compare & Contrast
Type of transformation
Reflection symmetry
Rotational symmetry
Reflection symmetry involves flipping a shape over a line to get a mirror image; rotational symmetry involves turning a shape around a central point.
Orientation of the line
Vertical line of symmetry
Horizontal line of symmetry
A vertical line of symmetry runs top to bottom (like in letter A); a horizontal line of symmetry runs left to right (like in letter H).
Exam Focus
Marks pattern: 2 MCQs (1 mark each) on identifying symmetry, 1 SA (2 marks) on drawing lines of symmetry, 1 LA (4 marks) on explaining rotational symmetry with an example
Most asked: Identifying lines of symmetry in letters and numbers · Identifying rotational symmetry in shapes and letters · Determining if a shape has both reflection and rotational symmetry
Don't skip: Which letters have both vertical and horizontal lines of symmetry?; Which digits have rotational symmetry?; Does the given design look the same after a 1/2 turn? Explain.
One-Line Revision (read aloud before the exam)
  1. Reflection symmetry means a shape can be divided into two mirror halves by a line of symmetry.
  2. The letter 'A' has one vertical line of symmetry.
  3. The letter 'H' has two lines of symmetry: vertical and horizontal.
  4. The letter 'H' also has rotational symmetry at a half turn.
  5. A firki has rotational symmetry because it looks the same after 1/4, 1/2, and 3/4 turns.
  6. A shape has rotational symmetry if it looks the same after a turn of less than a full turn.
  7. The letter 'O' has both vertical and horizontal lines of symmetry and rotational symmetry.
  8. The digit '0' has both reflection symmetry (vertical and horizontal) and rotational symmetry.
  9. The digit '8' has both reflection symmetry (vertical and horizontal) and rotational symmetry.
  10. The digit '1' has no reflection symmetry and no rotational symmetry.
  11. Block printing in Rajasthan uses wooden blocks with symmetrical designs.
  12. To check rotational symmetry, mark a dot on the shape and rotate it by 1/4, 1/2, and 3/4 turns.
  13. A full turn (360 degrees) does not count as rotational symmetry because every shape looks the same after a full turn.
  14. A shape can have only reflection symmetry, only rotational symmetry, or both.

MCQs — 10 Questions with Answers

Q1. Which alphabet letter has a vertical line of symmetry?

  • A. Letter E
  • B. Letter N
  • C. Letter A ✓
  • D. Letter K

Answer: C — Letter A has a vertical line of symmetry running down the middle, making the left and right halves mirror images.

Q2. How many quarter turns make one full turn?

  • A. 2 quarter turns
  • B. 3 quarter turns
  • C. 4 quarter turns ✓
  • D. 5 quarter turns

Answer: C — One full turn equals 4 quarter turns because each quarter turn is 1/4 of 360 degrees.

Q3. Which of the following has rotational symmetry at 1/4 turn?

  • A. A firki (windmill) ✓
  • B. A triangle
  • C. The letter C
  • D. A rectangle

Answer: A — A firki has rotational symmetry because it looks identical after spinning by 1/4, 1/2, and 3/4 turns.

Q4. To make a paper cutout of letter H by folding, how many times should you fold the paper?

  • A. Fold once
  • B. Fold twice (to make quarters) ✓
  • C. Fold three times
  • D. Do not fold

Answer: B — Letter H has two lines of symmetry, so you fold once vertically and once horizontally to make quarters, then draw 1/4 of the letter.

Q5. Which statement about reflection symmetry is correct?

  • A. Reflection symmetry means spinning a shape
  • B. Reflection symmetry means folding a shape so one half mirrors the other ✓
  • C. Reflection symmetry only works with triangles
  • D. Reflection symmetry needs a dot in the center

Answer: B — Reflection symmetry is when you fold a shape along a line and both halves look like mirror images of each other.

Q6. In the firki activity, after which turns does the firki look the same? (Fill in the blanks) The firki looks the same after _____, _____, _____, and a full turn.

  • A. 1/4, 1/2, 3/4 ✓
  • B. 1/3, 2/3, 1
  • C. 1/2, 3/4, 1
  • D. 1/8, 1/4, 1/2

Answer: A — A firki has 4-fold rotational symmetry, so it looks identical after 1/4 turn, 1/2 turn, 3/4 turn, and a full turn.

Q7. Which digit has both reflection symmetry AND rotational symmetry?

  • A. Digit 5
  • B. Digit 8 ✓
  • C. Digit 2
  • D. Digit 3

Answer: B — The digit 8 has a horizontal line of symmetry (reflection) and looks the same when rotated 1/2 turn (rotational symmetry).

Q8. Which of the following is an example of block printing with rotational symmetry?

  • A. A print that looks the same after 1/4 turn ✓
  • B. A print that is the same when folded
  • C. A print with only one line of symmetry
  • D. A print with no symmetry

Answer: A — Block printing designs with rotational symmetry repeat identically when the pattern is rotated by 1/4, 1/2, or 3/4 turns.

Q9. How many lines of symmetry does the letter O have?

  • A. 1 line
  • B. 2 lines
  • C. 4 lines
  • D. Infinite lines ✓

Answer: D — A circle (letter O) has infinite lines of symmetry because any line through its center creates a mirror image.

Q10. When you fold a square paper diagonally twice and cut along the X pattern for a firki, what do you create?

  • A. A shape with only reflection symmetry
  • B. A shape with only rotational symmetry
  • C. A firki that spins with rotational symmetry ✓
  • D. A shape with no symmetry

Answer: C — The firki activity creates a spinning toy with 4-fold rotational symmetry that looks identical every 1/4 turn.

Flashcards

What is reflection symmetry?

When a shape can be folded along a line so one half is an exact mirror image of the other half.

What is a line of symmetry?

The fold line where a shape looks the same on both sides as if reflected in a mirror.

How many lines of symmetry does letter A have?

Letter A has 1 vertical line of symmetry down the middle.

How many lines of symmetry does letter H have?

Letter H has 2 lines of symmetry: 1 vertical down the middle and 1 horizontal across the middle.

What is rotational symmetry?

When a shape looks the same after being rotated (spun) around its centre point by less than a full turn.

If a shape has rotational symmetry at 1/4 turn, what does this mean?

The shape looks identical when rotated by 1/4 of a full turn (90 degrees).

What is the relationship between quarter turn and half turn?

A half turn equals 2 quarter turns, and a full turn equals 4 quarter turns.

Which alphabet letter has both reflection and rotational symmetry?

Letter H has both reflection symmetry (vertical and horizontal lines) and rotational symmetry (at 1/2 turn).

How do you make an alphabet cutout using reflection symmetry?

Fold paper in half, draw half the letter along the fold, cut, then open to see the full letter.

What does a firki (windmill) demonstrate?

A firki demonstrates rotational symmetry because it looks the same after every 1/4 turn.

Important Board Questions

What is reflection symmetry? Give one example from the chapter. [1 mark]

Define reflection as folding so halves are mirror images; give any letter example like A, H, or O.

Draw a vertical line of symmetry on the letter H and explain why H has two lines of symmetry. [2 marks]

Draw letter H with a vertical line down the middle and a horizontal line across the middle; explain each makes a mirror image.

Explain how you would make a paper cutout of letter A using the folding method. What would you do in each step? [3 marks]

Four steps: fold paper in half, draw half the letter A on the fold, cut along outline, open the paper. Explain why folding works (uses reflection symmetry).

A firki (windmill) has rotational symmetry. Describe what happens when you rotate the firki by 1/4 turn, 1/2 turn, 3/4 turn, and a full turn. Does it look the same each time? Explain why a firki has rotational symmetry. [5 marks]

Describe that firki looks identical at each rotation point due to its 4-fold symmetrical design; explain rotational symmetry means shape appears unchanged after spinning less than full turn.

True or False: A shape can have both reflection symmetry and rotational symmetry. Give one example from the chapter to support your answer. [2 marks]

Answer is True; give letter H as example (has reflection symmetry with vertical and horizontal lines, and rotational symmetry at 1/2 turn) OR digit 8.

Assertion-Reason Questions for Symmetry

Q1. Assertion (A):

The letter A can be cut using symmetry by folding the paper once and drawing half of the letter.

Reason (R):

The letter A has exactly one vertical line of symmetry.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the assertion is true. Because A has one line of symmetry, folding once and drawing half the letter works perfectly.

Q2. Assertion (A):

A firki (windmill) has rotational symmetry.

Reason (R):

The firki looks the same when you rotate it by 1/4 turn.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true, and the reason is a correct explanation. The firki's appearance remaining unchanged during rotation (specifically at 1/4 turn) is what gives it rotational symmetry.

Q3. Assertion (A):

The digit 8 has reflection symmetry.

Reason (R):

The digit 8 has both horizontal and vertical lines of symmetry.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true, but the reason doesn't directly explain the assertion. While 8 does have these lines of symmetry, simply having lines of symmetry is what defines reflection symmetry. The reason is too specific rather than explanatory.

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Case-Based Questions (CBQ) for Symmetry

Case Study:

Deepa's mother is decorating the house for Diwali using paper cutouts. She wants to make cutouts of the letters 'H' and 'O' for a greeting banner. She remembers that using symmetry helps save time and paper when making cutouts. The letter H has two lines of symmetry (one vertical and one horizontal), while the letter O also has multiple lines of symmetry. Deepa's mother decides to fold the paper differently for each letter to use symmetry efficiently.

Q1. How many times should Deepa's mother fold the paper to make the letter H using symmetry?

  • A. Once
  • B. Twice ✓
  • C. Three times
  • D. Four times

Answer: B

Q2. Why is it efficient to use symmetry when making paper cutouts?

  • A. It makes the letters look prettier
  • B. It saves paper and time by drawing only a part of the shape ✓
  • C. It makes the paper easier to cut
  • D. It prevents the paper from tearing

Answer: B

Q3. If the letter O has symmetry in all directions, what type of symmetry does it definitely have?

  • A. No symmetry
  • B. Only rotational symmetry
  • C. Both reflection and rotational symmetry ✓
  • D. Only reflection symmetry

Answer: C

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