Reflection symmetry and lines of symmetry
A butterfly's two wings look like mirror pictures of each other. Fold the picture down the middle and the wings fall exactly on top of each other. That middle fold is a line of symmetry.
A shape has reflection symmetry if a fold line (the line of symmetry) makes the two halves match exactly, like a mirror image. A shape may have many lines, one line, or none.
Formula: Mirror image: same distance from the mirror line, on the other side. Distance dot to image = 2 × distance to the line.
In the figure: A square has 4 lines of symmetry, a rectangle (not a square) has 2, an equilateral triangle has 3.
Rule 1: Find lines of symmetry — Imagine folding the shape. Every fold that makes the halves match is a line of symmetry. Try vertical, horizontal and slanting folds.
- Easy: Square: 2 through side middles + 2 diagonals = 4 lines.
- Medium: Isosceles triangle: 1 line, from the top to the middle of the base.
- Exam-type: Scalene triangle: 0 lines.
Rule 2: Mirror images — A mirror line swaps left and right (vertical mirror) or top and bottom (horizontal mirror). Each point is as far behind the mirror as it is in front.
- Easy: A dot 3 squares left of the line has its image 3 squares right of the line.
- Medium: Distance between a dot 4 squares from the line and its image: 8 squares.
- Exam-type: b in a vertical mirror looks like d; in a horizontal mirror like p.
A line of symmetry gives two halves that are exact mirror images. Not every line through a shape is a line of symmetry.
Calling any dividing line a line of symmetry (a rectangle's diagonal is NOT one), or forgetting the diagonals of a square.
Fold in your mind: if the halves match exactly, it is a line of symmetry. Memory clue: "Fold and match — that's a line of symmetry."
Spot the difference
| Shape | Lines of symmetry |
|---|---|
| Square | 4 |
| Rectangle (not square) | 2 |
| Equilateral triangle | 3 |
| Scalene triangle | 0 |
Practice questions
- 1How many lines of symmetry does a square have?
- 2How many lines of symmetry does the letter H have?
- 3How many lines of symmetry does a scalene triangle have?
- 4A dot is 6 squares left of a vertical mirror line. Where is its image?
- 5The letter b is placed in front of a vertical mirror. Which letter is seen?
- 6A dot is 5 squares from a mirror line. How far is it from its image?
- 7How many lines of symmetry does an equilateral triangle have? Where do they go?
- 8How many lines of symmetry does a regular hexagon have?
Show answers (try them yourself first)
- 14
- 22
- 30
- 46 squares right of the line
- 5d
- 610 squares
- 73, each from a corner to the middle of the opposite side
- 86
Letters, digits and words with symmetry
When Prem and Manu cut out HAPPY BIRTHDAY letters, they folded the paper only for letters with a line of symmetry: draw half, cut, open, and the whole letter appears.
Capital letters sort into groups: a vertical line, a horizontal line, both, a half turn only, or none. Digits and whole words can have symmetry too.
Formula: Vertical: A M T U V W Y. Horizontal: B C D E K. Both: H I O X. Half turn only: N S Z. None: F G J L P Q R.
In the figure: Letters grouped by the symmetry they have (in the usual block style).
Rule 1: Test a letter — Fold it down the middle and across the middle, then turn it a half turn. Note what matches.
- Easy: H: both folds match and a half turn too.
- Medium: E: only the horizontal fold matches.
- Exam-type: N: no fold matches, but a half turn does.
Rule 2: Digits and words — Digits 0 and 8 have both reflection and half-turn symmetry; 3 has a horizontal line; 6 turns into 9. A word is symmetric in a mirror when every letter is symmetric and the word reads the same backwards.
- Easy: MOM and WOW look the same in a vertical mirror.
- Medium: BOX and CODE look the same in a horizontal mirror below.
- Exam-type: SOS looks the same after a half turn.
Only the usual block-style capital letters are meant here; a fancy font can change the answer.
Thinking N or S has a line of symmetry (they only have half-turn symmetry), or that M looks the same after a half turn (it becomes W).
Test each letter three ways: vertical fold, horizontal fold, half turn. Memory clue: "HIOX has it all; NSZ only turn."
Spot the difference
| Letter | Vertical | Horizontal | Half turn |
|---|---|---|---|
| H | Yes | Yes | Yes |
| A | Yes | No | No |
| E | No | Yes | No |
| N | No | No | Yes |
Practice questions
- 1Does the letter A have a horizontal line of symmetry?
- 2Which has a vertical line of symmetry: M or N?
- 3Name the letters that have both vertical and horizontal lines.
- 4Does E have a vertical line of symmetry?
- 5Which word looks the same in a vertical mirror: MOM or MAD?
- 6Which digit looks like 9 after a half turn?
- 7Does BOX look the same in a horizontal mirror below it?
- 8Does SOS look the same after a half turn?
Show answers (try them yourself first)
- 1No
- 2M
- 3H, I, O, X
- 4No, only a horizontal line
- 5MOM
- 66
- 7Yes, B, O and X all have horizontal lines
- 8Yes
Rotational symmetry: quarter, half and three-quarter turns
A firki (paper windmill) with 4 equal blades looks the same after each quarter turn. Turning an object about its centre and finding it looks unchanged is rotational symmetry.
A shape has rotational symmetry if it looks the same after turning about its centre by less than a full turn. Turns: quarter (1/4, 90°), half (1/2, 180°), three-quarter (3/4, 270°), full (360°).
Formula: 4 quarter turns = 2 half turns = 1 full turn = 360°. Clockwise order: north → east → south → west.
In the figure: A square with a marked dot after 1/4, 1/2, 3/4 and a full turn. The square looks the same each time, only the dot shows the turn.
Rule 1: Name the turn — Count how many quarter turns were made. Clock hands give a quick picture: 15 minutes = 1/4 turn, 30 = 1/2, 45 = 3/4, 60 = full.
Rotational symmetry: quarter, half and three-quarter turns
In the app you can move this figure with your finger.
Open in the app- Easy: Facing north and turning a half turn clockwise: facing south.
- Medium: Facing west and turning 3/4 clockwise: facing south.
- Exam-type: Minute hand in 45 minutes: 3/4 turn = 270°.
Rule 2: Does it look the same? — Mark a dot on the shape, turn it about the centre and check if the shape lands exactly on its old outline.
- Easy: H, N, S, Z and X look the same after a half turn.
- Medium: M turned a half turn looks like W: not the same.
- Exam-type: b turned a half turn looks like q.
A shape always looks the same after a full turn; rotational symmetry means it also matches after a smaller turn.
Mixing up clockwise and anticlockwise, or counting the full turn as a rotational symmetry by itself.
Put a dot on one corner and watch where the dot goes. Memory clue: "Quarter 90, half 180, three-quarter 270."
Spot the difference
| Turn | Fraction | Degrees |
|---|---|---|
| Quarter | 1/4 | 90° |
| Half | 1/2 | 180° |
| Three-quarter | 3/4 | 270° |
| Full | 1 | 360° |
Practice questions
- 1How many quarter turns make a full turn?
- 2A half turn is how many degrees?
- 3Facing east, turn a half turn clockwise. Which way now?
- 4The minute hand moves for 15 minutes. What turn is that?
- 5The letter b is given a half turn. What does it look like?
- 6The letter M is given a half turn. What does it look like?
- 7Facing south, turn 3/4 clockwise. Which way now?
- 8The hour hand moves from 12 to 6 and then from 6 to 9. What total turn is that?
Show answers (try them yourself first)
- 14
- 2180°
- 3West
- 41/4 turn
- 5q
- 6W
- 7East
- 83/4 turn
How many times a shape looks the same in a full turn
A ceiling fan with 3 equal blades looks the same 3 times as it makes one full turn. A cycle wheel with 8 equal spokes looks the same 8 times.
The order of rotational symmetry is the number of times a shape looks the same in one full turn. For a regular shape with n equal sides, the order is n and the smallest turn is 1/n of a full turn (360° ÷ n).
Formula: Smallest turn = 360° ÷ order. Regular n-sided shape: order n and n lines of symmetry.
In the figure: A triangle, square, pentagon and hexagon look the same 3, 4, 5 and 6 times in one full turn.
Rule 1: Count equal parts — Count the equal parts around the centre. That number is the order.
How many times a shape looks the same in a full turn
In the app you can move this figure with your finger.
Open in the app- Easy: Fan with 3 equal blades: order 3, smallest turn 1/3.
- Medium: Plus sign (+): order 4, smallest turn 1/4.
- Exam-type: Rectangle (not a square): order 2, smallest turn 1/2.
Rule 2: Smallest turn from the order — Divide 360° by the order, or write 1 divided by the order as a fraction of a turn.
- Easy: Order 6: 360° ÷ 6 = 60°, which is 1/6 of a turn.
- Medium: Order 8: 360° ÷ 8 = 45°, which is 1/8 of a turn.
- Exam-type: Order 12: 360° ÷ 12 = 30°.
The last match is the full turn, so it is counted in the order.
Counting only the matches before the full turn (square order is 4, not 3), or mixing up lines of symmetry and order for a rectangle (2 and 2) and a parallelogram (0 and 2).
Count equal parts around the centre; smallest turn = 360° ÷ that number. Memory clue: "Equal parts around the middle = order."
Spot the difference
| Shape | Lines | Order |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Parallelogram | 0 | 2 |
| Regular hexagon | 6 | 6 |
Practice questions
- 1How many times does a rectangle (not a square) look the same in a full turn?
- 2How many times does a square look the same in a full turn?
- 3How many times does a regular hexagon look the same in a full turn?
- 4An equilateral triangle first looks the same after which smallest turn?
- 5A fan has 3 equal blades. How many times does it look the same in a full turn?
- 6A shape looks the same after every 1/6 turn. How many times in a full turn?
- 7A shape looks the same 8 times in a full turn. What is the smallest turn?
- 8A rangoli matches itself 12 times in a full turn. What is the smallest turn in degrees?
Show answers (try them yourself first)
- 12
- 24
- 36
- 41/3 turn
- 53
- 66
- 71/8 turn (45°)
- 830°
Shapes with both, one or no symmetry
A kite flying in the sky is balanced left and right, but a turned kite does not look the same. A windmill toy looks the same when turned, but folding it does not match. Shapes can have either symmetry, both, or none.
Check two things for every shape: (1) any line of symmetry? (2) any turn smaller than a full turn that matches? The answers sort shapes into four groups: both, only reflection, only rotational, neither.
Formula: Both: square, rectangle, rhombus, equilateral triangle, regular hexagon. Only reflection: kite, isosceles triangle. Only rotational: parallelogram. Neither: scalene triangle.
In the figure: The same shapes have lines of symmetry and also turn onto themselves, so they have both kinds of symmetry (square and rectangle). An equilateral triangle also has both.
Rule 1: Two questions for every shape — First look for a fold line. Then look for a turn smaller than a full turn. Write Yes or No for each.
- Easy: Square: lines yes, turn yes: both.
- Medium: Isosceles triangle: line yes, turn no: only reflection.
- Exam-type: Parallelogram: line no, turn yes (half turn): only rotational.
Rule 2: Letters follow the same rule — H and O have both; A and M only reflection; N and S only rotational; F and R neither.
- Easy: H: both.
- Medium: N: only rotational.
- Exam-type: F: neither.
Rectangles and rhombuses (not squares) have 2 lines and order 2; a parallelogram has no line but order 2.
Saying a parallelogram has lines of symmetry because it looks balanced, or that a kite looks the same after a half turn.
Ask: fold? turn? Then pick the group. Memory clue: "Fold for a line, turn for a round."
Spot the difference
| Shape | Line? | Turn? |
|---|---|---|
| Square | Yes | Yes |
| Kite | Yes | No |
| Parallelogram | No | Yes |
| Scalene triangle | No | No |
Practice questions
- 1What symmetry does a kite have?
- 2What symmetry does a parallelogram have?
- 3What symmetry does the letter F have?
- 4What symmetry does a square have?
- 5What symmetry does an isosceles triangle have?
- 6What symmetry does the letter N have?
- 7Name one letter that has both kinds of symmetry.
- 8A rhombus (not a square): how many lines of symmetry and what order?
Show answers (try them yourself first)
- 1Only reflection symmetry
- 2Only rotational symmetry
- 3None
- 4Both
- 5Only reflection symmetry
- 6Only rotational symmetry
- 7H (or I, O, X)
- 82 lines, order 2
Paper cutouts, patterns and block prints
Rajasthan's artisans carve a design on a wooden block and print it again and again on cloth. Folded paper cutouts work the same way: one cut makes many matching copies.
Folding paper creates layers; one cut or punch goes through all layers. Each fold doubles the layers: 1 fold = 2, 2 folds = 4, 3 folds = 8. Printing a block around a centre, turned by equal turns, makes a pattern with rotational symmetry.
Formula: Layers after k folds = 2 × 2 × … (k times). Holes punched once = number of layers.
In the figure: Top: a punch through folded paper gives 2 holes after one fold and 4 after two folds. Bottom: 2 triangles make a rhombus, 6 make a hexagon, 4 squares make a big square.
Rule 1: Fold, cut, open — Fold on lines of symmetry. A cut on the fold itself joins into one shape; a cut on a free edge repeats once for each layer.
- Easy: Fold twice, punch once: 4 holes.
- Medium: Fold once, cut a half circle on the fold: 1 full circle.
- Exam-type: Fold twice, cut a triangle notch from the free edge: 4 notches.
Rule 2: Build designs from shapes — Join equal squares or equilateral triangles and test each new design for lines and turns.
- Easy: 2 triangles joined on a side: rhombus, 2 lines, order 2.
- Medium: 6 triangles around a point: hexagon, 6 lines, order 6.
- Exam-type: A block printed 4 times, each a quarter turn apart: order 4.
Fold only along lines of symmetry. Always count the layers before predicting the number of holes.
Saying 2 folds make 2 layers (they make 4), or expecting a cut on the fold to give two separate pieces.
Layers: 2, 4, 8, … — then holes = layers. Memory clue: "Each fold doubles."
Spot the difference
| Folds | Layers | Holes from one punch |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 4 | 4 |
| 3 | 8 | 8 |
Practice questions
- 1A paper is folded once and punched once away from the fold. How many holes?
- 2Folded twice and punched once. How many holes?
- 3Folded three times and punched once. How many holes?
- 4To cut the letter H, what part of the letter is drawn on the folded paper?
- 5A half circle is cut on the fold of a half-folded paper. What shape is the hole?
- 6A square is folded along both diagonals. How many triangles do the creases make?
- 76 equilateral triangles are joined around one point. What shape is made and what is its order?
- 8Folded twice, a triangle notch is cut from the free edge. How many notches when opened?
Show answers (try them yourself first)
- 12
- 24
- 38
- 4One-fourth (1/4)
- 5A full circle
- 64
- 7Hexagon, order 6
- 84