Symmetry — Balance and Beauty in Geometry
"Symmetry is everywhere — in a butterfly's wings, a snowflake, or the petals of a flower. It is the BALANCE that makes things beautiful."
1. What Is Symmetry?
A figure has symmetry if it can be divided or rotated such that its parts MATCH PERFECTLY.
Two types of symmetry in Class 7:
- Line symmetry (reflection/mirror symmetry) — folding along a line
- Rotational symmetry — turning about a point
2. Line Symmetry (Mirror Symmetry)
A figure has line symmetry if it can be folded along a line (the line of symmetry) so that the two halves MATCH EXACTLY.
'Imagine folding a piece of paper with a shape drawn on it. If the two halves align perfectly, the fold line is a line of symmetry.'
Key Properties
- The line of symmetry acts as a MIRROR — one half is the REFLECTION of the other.
- A shape may have ONE, MANY, or NO lines of symmetry.
- The line of symmetry can be VERTICAL, HORIZONTAL, or DIAGONAL.
Lines of Symmetry in Geometric Shapes
| Shape | Number of Lines | Diagram |
|---|---|---|
| Square | 4 | Two diagonals + one vertical + one horizontal through centre |
| Rectangle | 2 | One vertical + one horizontal through centre |
| Equilateral Triangle | 3 | From each vertex to the midpoint of the opposite side |
| Isosceles Triangle | 1 | From apex to the midpoint of base |
| Scalene Triangle | 0 | No line divides it into mirror halves |
| Circle | INFINITE | Every line through the centre |
| Regular Hexagon | 6 | 3 through opposite vertices + 3 through opposite midpoints |
| Parallelogram | 0 | (Not a rectangle/rhombus) — no line symmetry |
| Rhombus | 2 | Two diagonals |
| Isosceles Trapezium | 1 | Vertical through centre |
Line Symmetry in Letters
| Letter(s) | Lines of Symmetry |
|---|---|
| H, I, O, X | 2 (horizontal and vertical) |
| A, M, T, U, V, W, Y | 1 (vertical) |
| B, C, D, E, K | 1 (horizontal) |
| F, G, J, L, P, Q, R | 0 (no symmetry) |
3. Rotational Symmetry
A figure has rotational symmetry if it looks the SAME after being rotated (turned) by an angle LESS than 360° about a central point.
Centre of rotation: The fixed point about which the figure turns.
Key Concepts
Angle of rotation: The SMALLEST angle through which a figure can be rotated to look the SAME as the original.
Order of rotational symmetry: The number of times a figure matches its original position in ONE FULL rotation (360°).
Formula: Order = 360° ÷ Angle of rotation.
Rotational Symmetry of Common Shapes
| Shape | Angle of Rotation | Order of Symmetry |
|---|---|---|
| Equilateral Triangle | 120° | 3 |
| Square | 90° | 4 |
| Regular Pentagon | 72° | 5 |
| Regular Hexagon | 60° | 6 |
| Circle | Any angle | INFINITE |
| Rectangle | 180° | 2 |
| Parallelogram | 180° | 2 |
| Rhombus | 180° | 2 |
| Isosceles Triangle (non-equilateral) | 360° | 1 |
Checking Rotational Symmetry
'Trace the shape on paper. Mark the centre. Rotate the paper slowly. Each time the shape matches the original, that's one position.'
Example: A SQUARE.
- When rotated by 90°, it matches the original.
- When rotated by 180°, it matches again.
- 270° and 360° also match.
- Order = 4 (it matches 4 times in a full rotation).
4. Rotational Symmetry in Daily Life
| Object | Order | Reason |
|---|---|---|
| Fan blades (3-blade) | 3 | Matches every 120° |
| Windmill (4-blade) | 4 | Matches every 90° |
| Bicycle wheel spokes | Depends on number of spokes | 12 spokes → order 12 |
| Ceiling rose pattern | Typically 4 or 6 | Depends on design |
| Flower (5 petals) | 5 | Matches every 72° |
AP Context: 'The Konaseema Temple gopurams often have SYMMETRIC designs — both line symmetric and rotationally symmetric patterns in their architecture.'
5. Difference Between Line and Rotational Symmetry
| Aspect | Line Symmetry | Rotational Symmetry |
|---|---|---|
| Transformation | Folding (reflection) | Turning (rotation) |
| Axis | Line of symmetry | Point (centre of rotation) |
| Result | Two mirror halves | Same orientation after rotation |
| A figure can have | 0 to infinite lines | Order 1 to infinite |
| Examples | Butterfly, alphabet 'A' | Windmill, recycling symbol |
6. Figures with Both Types of Symmetry
| Figure | Lines of Symmetry | Rotational Order |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Equilateral triangle | 3 | 3 |
| Regular hexagon | 6 | 6 |
| Circle | Infinite | Infinite |
'Regular polygons have EQUAL number of lines of symmetry and order of rotational symmetry. This is a property of REGULAR figures — they are BOTH line-symmetric and rotationally symmetric to the same degree.'
7. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| A parallelogram has NO symmetry | It HAS rotational symmetry of order 2 | Parallelogram has NO line symmetry but DOES have rotational symmetry (180°) |
| A scalene triangle has 1 line of symmetry | No line divides it into two equal halves | Scalene triangles have NO line symmetry |
| Rotational symmetry of a rectangle is 4 | Only matches at 180° and 360° | Only 2 matches — order 2 |
| Every shape that has line symmetry also has rotational symmetry | NOT true | Isosceles triangle has line symmetry (1 line) but NO rotational symmetry (order 1) |
| A line of symmetry goes through the centre ONLY | Not always — isosceles triangle's line goes through apex and base | It goes through the centre of the figure usually, but the exact position depends |
8. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Identifying lines of symmetry | 2-3 | Count lines in given shapes |
| Rotational symmetry — order and angle | 3-4 | Find order for given shapes |
| Both symmetries combined | 2-3 | Compare and contrast |
| Symmetry in real life | 1-2 | Give examples |
Quick Self-Test
Q1. How many lines of symmetry does a regular pentagon have? A1. 5.
Q2. What is the order of rotational symmetry of an equilateral triangle? A2. 3.
Q3. Does a parallelogram have line symmetry? A3. No. A parallelogram has NO line symmetry (unless it is a rectangle or rhombus).
Q4. What is the angle of rotation of a square? A4. 90°.
Q5. Give an example of a figure with NO line symmetry but WITH rotational symmetry. A5. A parallelogram — no line symmetry, but rotational symmetry of order 2 (180°).
Q6. A shape matches its original 5 times in a full 360° rotation. What is its order of rotational symmetry and angle of rotation? A6. Order = 5. Angle of rotation = 360°/5 = 72°.
Q7. Which has more lines of symmetry: a square or a rectangle? A7. A square has 4 lines of symmetry, a rectangle has 2. Square has more.
