1. Chapter at a Glance
Look at a butterfly, the Taj Mahal, a rangoli or the Ashoka Chakra — they look balanced and beautiful because they have symmetry. A shape is symmetrical when one part matches another part exactly, either by folding (reflecting) or by turning (rotating). In this chapter you will learn to find lines of symmetry, understand rotational symmetry, measure its angle and order, and spot symmetry in letters, shapes, nature and art.
- Line (reflection) symmetry and lines of symmetry.
- Figures with no, one, two or many lines of symmetry.
- Rotational symmetry, the angle of symmetry and the order.
- Symmetry of regular polygons, letters and everyday objects.
- Creating symmetrical designs.
2. Line Symmetry
A figure has line symmetry if a line can divide it into two parts that match exactly when the figure is folded along that line. The folding line is called the line of symmetry or axis of symmetry. Each half is the mirror image (reflection) of the other.
- Fold-test: fold the paper along the line; if the two halves cover each other completely, the line is a line of symmetry.
- Mirror-test: place a mirror along the line; if the reflection completes the figure correctly, it is a line of symmetry.
- Ink-blot test: put ink on one half of a sheet and fold — the blot on the other half is the mirror image.
Number of lines of symmetry
| Figure | Lines of symmetry |
|---|---|
| Scalene triangle | 0 |
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Rectangle (not a square) | 2 |
| Square | 4 |
| Rhombus (not a square) | 2 (its diagonals) |
| Parallelogram (general) | 0 |
| Kite | 1 |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Circle | infinitely many (every diameter) |
A regular polygon with n sides has exactly n lines of symmetry.
The diagonal of a rectangle is not a line of symmetry. If you fold a rectangle along its diagonal, the two triangles do not cover each other — the corners stick out. Only in a square do the diagonals act as lines of symmetry.
Symmetry in letters and numbers
- 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.
- Both vertical and horizontal: H, I, O, X.
- No line of symmetry: F, G, J, L, N, P, Q, R, S, Z.
- Digits: 0, 1, 3 and 8 have lines of symmetry (in usual print styles); 2, 4, 5, 6, 7 do not.
3. Rotational Symmetry
A figure has rotational symmetry if it looks exactly the same after being turned about a fixed point (its centre) by some angle less than a full turn (360°).
- The fixed point is the centre of rotation.
- The smallest angle by which the figure can be turned to look the same is the angle of rotational symmetry.
- The number of times the figure looks the same during one full turn is the order of rotational symmetry.
| Figure | Angle of symmetry | Order |
|---|---|---|
| Equilateral triangle | 120° | 3 |
| Square | 90° | 4 |
| Rectangle | 180° | 2 |
| Regular pentagon | 72° | 5 |
| Regular hexagon | 60° | 6 |
| Letter S, Z, N | 180° | 2 |
| Circle | any angle | infinite |
A flower design looks the same after a turn of 45°. What is its order of rotational symmetry?
Solution: Order = 360° ÷ 45° = 8.
The Ashoka Chakra has 24 equally spaced spokes. Find its angle of rotational symmetry.
Solution: Angle = 360° ÷ 24 = 15°, and the order is 24.
4. Figures with Both Kinds of Symmetry
Many figures have both line symmetry and rotational symmetry, but the two are independent:
- Square, equilateral triangle, regular hexagon, circle, letter H: both line and rotational symmetry.
- Letters S, Z, N; a general parallelogram: rotational symmetry (order 2) but no line symmetry.
- Letter A, isosceles triangle, a kite: line symmetry but no rotational symmetry.
- Letter F, a scalene triangle: neither.
For a regular polygon with n sides: n lines of symmetry, angle of rotational symmetry 360° ÷ n, and order n.
5. Symmetry in Nature, Art and Daily Life
- Nature: leaves and butterflies (line symmetry), starfish (5 lines and order 5), snowflakes (6 lines and order 6), sunflower heads.
- Architecture: the Taj Mahal, temple gopurams and the Golconda and Hampi structures show strong line symmetry.
- Art: rangoli, kolam, mehndi patterns and block prints use both kinds of symmetry.
- Logos and wheels: car wheels, fans and the Ashoka Chakra use rotational symmetry.
Making symmetrical designs
- Draw a line of symmetry (or mark a centre of rotation).
- Draw a pattern on one side.
- Reflect each point to the other side at the same distance from the line (or turn the pattern by equal angles about the centre).
- Check by folding or turning a tracing.
6. Symmetry of Quadrilaterals Explained
Quadrilaterals are a good place to see why symmetry depends on equal sides and equal angles.
- Square: all sides and all angles equal, so it can be folded along both diagonals and along both lines joining midpoints of opposite sides — 4 lines. It also looks the same after every quarter turn.
- Rectangle: only opposite sides are equal, so only the two lines through the midpoints of opposite sides work — 2 lines; it looks the same after a half turn.
- Rhombus: all sides equal but angles not all equal, so it folds along its diagonals — 2 lines; order 2.
- Kite: two pairs of equal adjacent sides, so it folds only along the diagonal between the unequal angles — 1 line; no rotational symmetry.
- Isosceles trapezium: folds along the line joining the midpoints of the parallel sides — 1 line.
Complete a figure that is symmetric about a vertical line when its left half is a right triangle with the right angle on the line.
Solution: Reflect each corner of the triangle across the line to the same distance on the other side. The two right triangles together make an isosceles triangle, which indeed has exactly one line of symmetry — the given line.
7. Master Formula Table
| Fact | Rule | Example |
|---|---|---|
| Regular n-gon: lines of symmetry | n | Hexagon → 6 |
| Regular n-gon: angle of symmetry | 360° ÷ n | Pentagon → 72° |
| Order of rotation | 360° ÷ angle | Angle 40° → order 9 |
| Circle | infinite lines and order | Every diameter |
| No rotational symmetry | order 1 (only 360°) | Letter A |
8. Master Summary Table
| Figure | Lines of symmetry | Order of rotation |
|---|---|---|
| Scalene triangle | 0 | 1 |
| Isosceles triangle | 1 | 1 |
| Equilateral triangle | 3 | 3 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Parallelogram | 0 | 2 |
| Square | 4 | 4 |
| Regular hexagon | 6 | 6 |
| Circle | Infinite | Infinite |
Chart: lines of symmetry of regular polygons
9. Mind Map
- Fold or mirror test
- Mirror image halves
- Regular n-gon: n lines
- Centre of rotation
- Angle of symmetry
- Order = 360° ÷ angle
- Letters and digits
- Quadrilaterals
- Circle: infinite
- Nature, rangoli
- Monuments
- Wheels, Ashoka Chakra
10. Tips and Tricks
- Use tracing paper: trace the figure, then flip (for lines) or turn (for rotation) to test.
- For rotational symmetry, count how many identical "arms" or repeats a figure has — that is usually its order.
- A figure with two perpendicular lines of symmetry always has rotational symmetry of order at least 2.
- For regular polygons, number of sides = lines of symmetry = order of rotation.
- Counting the diagonals of a rectangle as lines of symmetry.
- Saying a parallelogram has line symmetry — a general parallelogram has none.
- Saying the order of a figure with no rotational symmetry is 0 (it is 1).
- Forgetting that the letter S has rotational symmetry of order 2.
11. Important One-liners
- A line of symmetry divides a figure into two mirror-image halves.
- A scalene triangle has no line of symmetry.
- An equilateral triangle has 3 lines of symmetry.
- A rectangle has 2 lines of symmetry; its diagonals are not lines of symmetry.
- A square has 4 lines of symmetry and rotational order 4.
- A circle has infinitely many lines of symmetry.
- Order of rotational symmetry = 360° ÷ smallest angle of turn.
- A regular hexagon has angle of symmetry 60° and order 6.
- The letters S, N and Z have rotational symmetry of order 2 but no line symmetry.
- The letters H, I, O and X have both horizontal and vertical lines of symmetry.
- The Ashoka Chakra, with 24 spokes, has an angle of symmetry of 15°.
12. Exam-pattern Questions with Detailed Answers
How many lines of symmetry does a regular pentagon have?
- 2
- 4
- 5
- 10
Show answer
A regular polygon with n sides has n lines of symmetry. For a pentagon, n = 5.
Answer: (C) 5
Which of these letters has no line of symmetry: A, E, R, T?
Show answer
A and T have vertical lines; E has a horizontal line. R has no line of symmetry.
Answer: R
What is the order of rotational symmetry of a square?
Show answer
A square looks the same after turns of 90°, 180°, 270° and 360°. Order = 360° ÷ 90° = 4.
Answer: 4
A figure has rotational symmetry with angle 30°. What is its order?
Show answer
Order = 360° ÷ 30° = 12.
Answer: 12
Name a quadrilateral that has rotational symmetry of order 2 but no line of symmetry.
- Square
- Rectangle
- Parallelogram (non-rectangular, non-rhombus)
- Kite
Show answer
A general parallelogram looks the same after a half turn (180°) about the point where its diagonals meet, but folding along any line does not match its halves.
Answer: (C) Parallelogram
Why is the diagonal of a rectangle (that is not a square) not a line of symmetry?
Show answer
When folded along a diagonal, the longer side falls along the shorter side and the corners do not meet; the two triangles are equal in area but are not mirror images along that line.
Answer: The halves do not coincide on folding
Which of the digits 0 to 9 have both a horizontal and a vertical line of symmetry (in simple straight-line digital style)?
Show answer
In seven-segment digital style, 0 and 8 have both a vertical and a horizontal line of symmetry. 1 has a vertical line only in some styles; 3 has only a horizontal line.
Answer: 0 and 8
A star shape has 5 identical points. State its lines of symmetry and order of rotation.
Show answer
A regular 5-pointed star has 5 lines of symmetry (one through each point) and rotational order 5 (angle 72°).
Answer: 5 lines; order 5
A regular polygon has an angle of rotational symmetry of 40°. How many sides and lines of symmetry does it have?
Show answer
Order = 360° ÷ 40° = 9. For a regular polygon, order = number of sides = number of lines of symmetry.
Answer: 9 sides and 9 lines of symmetry (a regular nonagon)
Can a figure have exactly 2 lines of symmetry but no rotational symmetry? Explain.
Show answer
No. Two lines of symmetry always meet at a point. Reflecting in one line and then in the other is the same as rotating about the meeting point by twice the angle between the lines. So the figure must also have rotational symmetry (for perpendicular lines, order 2 — like a rectangle).
Answer: No — two lines of symmetry force rotational symmetry
Through what angles (less than 360°) can a regular hexagon be turned so that it looks the same?
Show answer
Angle of symmetry = 360° ÷ 6 = 60°. Multiples less than 360°: 60°, 120°, 180°, 240°, 300°.
Answer: 60°, 120°, 180°, 240°, 300°
A rangoli has 8 identical petals arranged around a centre, each petal symmetrical about its own middle line. How many lines of symmetry and what order of rotation does the rangoli have?
Show answer
Rotating by 360° ÷ 8 = 45° moves each petal onto the next, so order = 8.
Lines of symmetry: one through each petal's middle (8 lines, but opposite petals share a line, giving 4) plus one between each pair of neighbouring petals (another 4). Total = 8.
Answer: 8 lines of symmetry; order 8
13. Quick Revision Summary
- Line symmetry: folding along the line makes the halves match (mirror images).
- Rotational symmetry: the figure looks the same after a turn less than 360°; order = 360° ÷ smallest angle.
- Regular n-gon: n lines of symmetry, angle 360°/n, order n; circle: infinite.
- Rectangle: 2 lines, order 2; square: 4 lines, order 4; parallelogram: 0 lines, order 2.
- Symmetry appears in nature, monuments, rangoli, wheels and the Ashoka Chakra.