Chapter 9 — Symmetry
Ganita Prakash | Grade 6 — Structured Visual Notes
When we look around us, many objects catch our eye — a flower, a butterfly, a rangoli, a pinwheel, and grand buildings like the Taj Mahal or a temple gopuram. What makes them pleasing is that some part of the figure repeats in a fixed, orderly pattern.
Symmetry means a part or parts of a figure repeat in some definite pattern. A figure that has this property is called symmetrical. A cloud has no repeating pattern, so it is not symmetrical.
In the chapter-opening pictures, the flower, butterfly, rangoli and pinwheel are all symmetrical, while the cloud is not.
9.1 Line of Symmetry
Imagine folding a shape along a straight line. If one half lands exactly on top of the other half and covers it completely, those two halves are called mirror halves.
A line that cuts a figure into two parts which overlap exactly when the figure is folded along it is called a line of symmetry (also called an axis of symmetry).
Note: a shape can be cut into two equal-looking pieces without those pieces overlapping. For example, four puzzle pieces split by a middle line are equal in count, but the halves do not fold onto each other — so that middle line is not a line of symmetry.
Figures with more than one line of symmetry
A square is a great example. If you take a square paper and keep folding it, you find it can be folded four different ways so that halves overlap: a vertical fold, a horizontal fold, and two diagonal folds. So a square has 4 lines of symmetry.
Reflection
Folding along a line of symmetry is the same idea as reflection. The part on one side of the line gets reflected to the other side, like a mirror image. A figure that has one or more lines of symmetry is said to have reflection symmetry.
Take a square with corners labelled A, B, C, D (A top-left, B top-right, C bottom-right, D bottom-left).
| Line of reflection | Where the corners go |
|---|---|
| Vertical line | B → A's place, C → D's place, A → B's place, D → C's place |
| Diagonal A to C | A and C stay fixed; B ↔ D swap places |
| Horizontal line | A ↔ D swap, B ↔ C swap |
Generating symmetric shapes
We can create symmetric shapes easily:
- Ink blot method: Fold paper in half, drop ink on one side, press and open. The blot becomes symmetric, and the fold line is its line of symmetry.
- Paper folding and cutting: Fold paper and cut along a line; when opened, the shape is symmetric about the fold.
- Punching game: Punch a hole in folded paper; on opening, the holes appear as mirror pairs about the fold line.
Figure it Out — Solutions (Section 9.1)
Along the horizontal line: D and C take the positions earlier held by A and B respectively.
Punching & Paper-cutting — Solutions (Page 224–226)
(b) Fold horizontally then vertically. At the closed corner, cut a slanting straight line. On opening, this gives a square (diamond) hole at the centre.
Imp: Always check that the 4-sided central figure has all equal sides and equal (right) angles to be a true square.
Lines of Symmetry of Common Shapes — Solutions (Q6, Q7)
| Shape | Number of Lines of Symmetry |
|---|---|
| Square (tilted) | 4 |
| 8-pointed star | 8 |
| Equilateral triangle (equal sides & angles) | 3 |
| Regular hexagon (equal sides & angles) | 6 |
b. Exactly three lines → an equilateral triangle.
c. No line → a scalene triangle.
No, it is not possible to draw a triangle with exactly two lines of symmetry. A triangle can have 0, 1 or 3 lines only.
b. Two lines → an oval / leaf-pair figure (vertical + horizontal).
c. Four lines → a four-petal (clover) flower shape.
9.2 Rotational Symmetry
A paper windmill looks orderly, yet it has no line of symmetry — folding it does not make the halves overlap. But if you rotate it by 90° about the centre point, it looks exactly the same as before. This is a new kind of symmetry.
A figure has rotational symmetry if it looks exactly the same after being rotated by some angle (more than 0° and less than 360°) about a fixed point.
The fixed point is the centre of rotation.
An angle that brings the figure onto itself is an angle of symmetry (angle of rotational symmetry).
For the windmill, the angles of symmetry are 90° (quarter turn), 180° (half turn), 270° (three-quarter turn) and 360° (full turn). So it has 4 angles of symmetry.
Square and rotation
A square comes back onto itself after 90°. This sends A→B, B→C, C→D, D→A. So a square has the same 4 angles of symmetry: 90°, 180°, 270°, 360°. The centre of rotation is the point where the diagonals meet.
Figures that do NOT have rotational symmetry
Consider a plain slanted strip (a parallelogram-like bar). Rotating it 180° does not match the original, and only a full 360° turn brings it back. So this figure does not have rotational symmetry — 360° is its only angle of symmetry.
Figures with radial arms
Radial arms are simple lines/spokes coming out from a centre. They help us build figures with any number of angles of symmetry.
If a radial-arm figure has n equally spaced arms, the angle between two adjacent arms must be 360° ÷ n. Then the figure has n angles of symmetry.
| Number of arms (n) | Angle between arms | Angles of symmetry |
|---|---|---|
| 2 | 180° | 180°, 360° |
| 3 | 120° | 120°, 240°, 360° |
| 4 | 90° | 90°, 180°, 270°, 360° |
| 5 | 72° | 72°, 144°, 216°, 288°, 360° |
| 6 | 60° | 60°, 120°, 180°, 240°, 300°, 360° |
For 3 arms: the three angles A, B, C must be equal, and together make 360°, so each is 360° ÷ 3 = 120°. Only then does the 3-arm figure have rotational symmetry.
The Pattern in Angles of Symmetry
- When there are exactly 2 angles → 180°, 360° (multiples of 180).
- When there are exactly 3 angles → 120°, 240°, 360° (multiples of 120).
- When there are exactly 4 angles → 90°, 180°, 270°, 360° (multiples of 90).
In every case, all angles of symmetry are multiples of the smallest angle of symmetry.
- Every figure has 360° as an angle of symmetry. → True
- If the smallest angle of symmetry is a whole number of degrees, then it is a factor of 360. → True
Symmetries of a Circle
The circle is the most symmetric shape. Rotating a circle about its centre by any angle leaves it looking the same — so every angle is an angle of symmetry, and there is no single "smallest" angle. Also, every diameter of a circle is a line of symmetry. Wheels, fans and flowers show similar rotational symmetry around us.
Figure it Out — Solutions (Section 9.2)
b. (two unequal end pieces) → only 360°.
c. (T-shaped / 2-fold) → 180°, 360°.
(b) X shape → order 4 (if arms equal) / 2.
(c) 6-pointed star → order 6.
(d) rotating arms figure → order 3.
(e) plus (cross) → order 4.
(f) regular pentagon → order 5.
(i) Colour every 4th sector to get 3 angles of symmetry.
(ii) Colour every 3rd sector to get 4 angles of symmetry.
(iii) Possible numbers of angles of symmetry from colouring the 12 sectors are the factors of 12: 1, 2, 3, 4, 6, 12 (up to 12 possible).
b. Isosceles triangle → 1 line, no rotational symmetry.
c. Parallelogram → no line of symmetry, but 2 angles (180°, 360°) of rotational symmetry.
d. Isosceles trapezium → 1 line of symmetry, but no rotational symmetry.
b. 17°? No — 360° is not a multiple of 17°.
b. Yes, it has rotational symmetry. Angles of rotational symmetry: 120°, 240°, 360°.
| Regular Polygon | Lines of Symmetry | Angles of Symmetry |
|---|---|---|
| Triangle | 3 | 3 |
| Quadrilateral (square) | 4 | 4 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 6 |
| Heptagon | 7 | 7 |
| Octagon | 8 | 8 |
| Nonagon | 9 | 9 |
| Decagon | 10 | 10 |
Number of angles of symmetry: 3, 6, 6, 6, 6 …
Imp Summary Flow
- Line of symmetry → two halves overlap on folding.
- Angle of symmetry → figure matches itself on rotating (between 0° and 360°).
- Square: 4 lines + 4 angles. Rectangle: 2 lines only. Equilateral triangle: 3 lines + 3 angles.
- Circle: infinite lines (every diameter) + every angle is an angle of symmetry.
- 360° is always an angle of symmetry; smallest whole-number angle is a factor of 360.
- A figure may have only lines, only angles, or both — the two symmetries are independent.