Symmetrical Designs
Reflection (mirror) symmetry and rotational (turning) symmetry in letters, digits, paper cutouts, firkis, and block prints.
- Lines of symmetry: vertical, horizontal, and slanting
- Making cutouts by folding paper in half or in quarters
- Rotational symmetry: when a shape looks the same after a ¼, ½, or ¾ turn
- Symmetry in letters, digits, rangoli, and block printing
🦋 Let’s Begin: The Butterfly Test
Meera put a blob of paint on one side of a paper, folded it, and pressed. When she opened it: a butterfly! Both halves were exactly the same, like a reflection in a mirror.
The fold line is called the line of symmetry.
🪞 Part 1: Reflection Symmetry
A shape has reflection symmetry if you can fold it so that one half sits exactly on the other half.
| Line of symmetry | Letters (capital) |
|---|---|
| Vertical only ( | ) | A, M, T, U, V, W, Y |
| Horizontal only ( — ) | B, C, D, E, K |
| Both vertical and horizontal | H, I, O, X |
| None | F, G, J, L, N, P, Q, R, S, Z |
(Some of these depend on how the letter is written, so always check your own letters.)
Paper cutouts
- One line of symmetry (like A or a diya): fold once, draw half, cut, and open.
- Two lines of symmetry (like H or a flower): fold twice, draw a quarter, cut, and open.
🌀 Part 2: Rotational Symmetry
Some shapes look the same even after you turn them (without flipping). This is rotational symmetry.
Think of a ceiling fan with 3 blades. Turn it by a third of a full turn, and it looks exactly the same. A 4-petal flower looks the same after every ¼ turn. A firki (paper windmill) looks the same after ¼, ½, and ¾ turns.
| Shape | Looks the same after | How many times in 1 full turn |
|---|---|---|
| Letter S, N, Z | ½ turn | 2 |
| Letter H, I | ½ turn | 2 |
| Square, firki, plus sign (+) | ¼, ½, ¾ turn | 4 |
| Letter L, F | Only a full turn | 1 (no rotational symmetry) |
✅ It must look the same before completing a full turn to have rotational symmetry.
Trick to check: Trace the shape on tracing paper, put a pin in the centre, and turn it slowly. Count how many times it matches before you complete a full turn.
🔢 Part 3: Symmetry in Numbers
- 0 and 8 have both reflection and rotational (½ turn) symmetry.
- 3 has a horizontal line of symmetry.
- 6 and 9 turn into each other with a ½ turn. So 69, 96, and 6009 look the same after a ½ turn!
Numbers like 88, 808, 8008, 1001 (with straight 1s) have both kinds of symmetry.
🎨 Part 4: Symmetry in Art
Rangoli, kolam, block prints from Rajasthan (Sanganer and Bagru), and phulkari embroidery all use symmetry. An artisan presses the same wooden block again and again, turning it each time, to make a design with rotational symmetry.
Example: Press a leaf-shaped block 4 times, turning it by ¼ each time around one centre point. You get a 4-leaf design that looks the same after every ¼ turn.
❌ Common Mistakes to Avoid
✅ The two halves must match exactly when folded.
✅ For rotational symmetry, only turn the shape.
✅ It has 2. A square has 4.
✅ In coloured designs, colours must also match.
✏️ Practice Time
Try these on your own first, then check your answer below each question.
📘 NCERT Textbook Activity Help
Hints and answers for the activities in the textbook, page by page. Try each one yourself before looking at the answer!
Page 136 · Alphabet CutoutHalf or quarter cutouts
Horizontal line of symmetry: E, X, K, O
Vertical line of symmetry: X, T, V, O
Both: X, O
Does it look the same?
Rotational symmetry in letters and digits
Digits (as drawn in the book): reflection symmetry: 0, 1, 3, 8 · rotational symmetry: 0, 1, 8 · both: 0, 1, 8
|| and |00|: both symmetries (c).
Examples: 2-digit: 11, 88 (both), 69 (rotational only), 33 (reflection, horizontal line) · 3-digit: 101, 808 (both), 619 (rotational) · 4-digit: 1001, 8008 (both), 6009 (rotational)
Adding shapes for symmetry
(b) Add a circle on the left side, next to the left triangle. Now it looks the same after a ½ turn.
(c) Make all four arms the same: a triangle and a circle on every side (top, bottom, left, and right). Now it looks the same after a ¼ turn.
(d) Yes, the new designs have reflection symmetry: vertical and horizontal lines (and diagonal lines for the ¼ turn design).
Checkerboard square
To look the same after every ¼ turn: colour the 8 triangles alternately (colour 1, colour 2, colour 1, colour 2 … going around). It looks the same 4 times in a full turn.
Reflection: the checkerboard has 2 diagonal lines of symmetry. The alternating-triangle design has no lines of symmetry. (If you colour it in matching pairs, it can have some.)
Symmetry in the blue-and-yellow shape
Match the blocks and prints
Design B looks the same after every ¼ turn. It has rotational symmetry (and also reflection symmetry, along the dotted lines).
Border shapes: regular shapes (triangle, square, pentagon, hexagon), circles, and stars have both reflection ✓ and rotational * symmetry. A non-square rectangle and a rhombus have 2 lines and a ½-turn symmetry. A parallelogram has only rotational symmetry.
👨👩👧 Tips for Parents
- Make paper cutouts together: fold, draw half, cut, and open.
- Look at rangoli, bangles, and fabric prints, and ask, “Where is the line of symmetry?”
- Use a small mirror on half a picture to see the full shape.