Line of symmetry and mirror halves
Look at a butterfly: its left wing matches its right wing. A child folds a sheet in half and cuts a paper flower, and when opened both sides are the same. In such shapes one half is the mirror picture of the other.
A figure has a line of symmetry if folding it along that line makes the two halves fit exactly on each other. The two halves are called mirror halves. The line is also called the mirror line, and it can be upright, flat or slanting.
In the figure: The triangle on the left folds into matching halves; the one on the right does not.
Rule 1: The fold test — Imagine folding along the line. If every point of one half lands exactly on a point of the other half, the line is a line of symmetry. If any part sticks out, it is not.
Line of symmetry and mirror halves
In the app you can move this figure with your finger.
Open in the app- Easy: An isosceles triangle folded along the line from its top corner to the middle of its base: the halves fit, so it is a line of symmetry.
- Medium: A rectangle folded along the line joining the middle points of its two longer sides: the halves fit.
- Exam-type: A triangle with all three sides different: no fold makes the halves fit, so it has no line of symmetry.
Rule 2: Matching points — Every point has a partner on the other side of the line, at the same distance from the line. A point on the line is its own partner. So the distance between a point and its partner is twice its distance from the line.
- Easy: A point 3 cm left of the line has a partner 3 cm right of it, so they are 6 cm apart.
- Medium: A point 5 cm from the line is 5 × 2 = 10 cm from its partner.
- Exam-type: The capital letter A has one upright line of symmetry, H has two (upright and flat), and F has none.
A line that cuts a figure into two equal-sized pieces is not always a line of symmetry; the pieces must also fit by folding.
Taking the diagonal of a rectangle (that is not a square) as a line of symmetry; folding along it leaves the corners sticking out.
Fold it: if the halves match exactly, the fold line is a line of symmetry. Memory clue: “Fold, Fit, Finished – a line of symmetry”.
Spot the difference
| Figure | Fold along | Halves fit? |
|---|---|---|
| Isosceles triangle | top corner to base middle | yes |
| Rectangle | middle of long sides joined | yes |
| Rectangle | a diagonal | no |
| Scalene triangle | any line | no |
Practice questions
- 1What do we call a line that folds a figure into two matching halves?
- 2Does a triangle with three different sides have a line of symmetry?
- 3A point is 4 cm from the line of symmetry. How far from the line is its partner point?
- 4A point P is 6 cm from the line of symmetry. What is the distance between P and its partner point?
- 5A rectangle is 8 cm by 5 cm. Is its diagonal a line of symmetry?
- 6How many lines of symmetry does an isosceles triangle have?
- 7Of the capital letters A, H, F and L, how many have at least one line of symmetry?
- 8From the line of symmetry to the left edge of a figure is 9 cm. What is the full width of the figure?
Show answers (try them yourself first)
- 11) A line of symmetry
- 22) No
- 33) 4 cm
- 44) 12 cm
- 55) No
- 66) 1
- 77) 2 (A and H)
- 88) 18 cm
Counting lines of symmetry
A square floor tile, a triangular warning sign and a round plate can each be folded in different ways to match. A designer counts how many such folds a pattern has before choosing it.
Different shapes have different numbers of lines of symmetry. A regular polygon has as many as it has sides. A circle has countless lines, one for every diameter. Some shapes, such as a triangle with three different sides, have none.
In the figure: Triangle 3, rectangle 2, square 4, hexagon 6 lines of symmetry.
Rule 1: Regular polygons: as many as the sides — A regular polygon has as many lines of symmetry as sides. All the lines pass through the centre, and two neighbouring lines make an angle of 180° divided by the number of sides.
Counting lines of symmetry
In the app you can move this figure with your finger.
Open in the app- Easy: Regular pentagon: 5 sides, so 5 lines of symmetry.
- Medium: Regular hexagon: 6 lines, and neighbouring lines are 180 ÷ 6 = 30° apart.
- Exam-type: Regular octagon: 8 lines of symmetry.
Rule 2: Triangles and four-sided shapes — A triangle with three equal sides has 3 lines, with two equal sides has 1, and with no equal sides has 0. A rectangle that is not a square has 2 lines, a square 4, a kite 1 and a parallelogram that is not a rectangle has 0.
- Easy: Rectangle 9 cm by 5 cm: 2 lines, the ones joining the middle points of opposite sides (its diagonals do not count).
- Medium: Isosceles triangle: 1 line.
- Exam-type: Slanting parallelogram: 0 lines, however you fold it.
A circle has endless lines of symmetry: every line through its centre is one.
Counting the diagonals of a rectangle (that is not a square) as lines of symmetry.
Regular polygon: number of lines = number of sides. Memory clue: “Regular shape: lines equal sides”.
Spot the difference
| Shape | Lines of symmetry | Why |
|---|---|---|
| Equilateral triangle | 3 | one through each corner |
| Rectangle | 2 | two middle lines |
| Square | 4 | 2 middle lines + 2 diagonals |
| Circle | countless | every line through the centre |
Practice questions
- 1How many lines of symmetry does a square have?
- 2How many lines of symmetry does a regular hexagon have?
- 3How many lines of symmetry does a rectangle that is not a square have?
- 4A regular polygon has 9 lines of symmetry. How many sides does it have?
- 5What angle do neighbouring lines of symmetry of a regular hexagon make?
- 6Which has more lines of symmetry, a square or a regular pentagon?
- 7Neighbouring lines of symmetry of a regular polygon make an angle of 20°. How many sides does it have?
- 8How many lines of symmetry do a square, a rectangle (not a square) and an equilateral triangle have altogether?
Show answers (try them yourself first)
- 11) 4
- 22) 6
- 33) 2
- 44) 9
- 55) 30° (180 ÷ 6)
- 66) The pentagon (5 lines against 4)
- 77) 9 sides (180 ÷ 20)
- 88) 9 (4 + 2 + 3)
Rotational symmetry and centre of rotation
A ceiling fan with three equal blades looks the same after it turns one-third of a round. A bicycle wheel with evenly spaced spokes does the same. These shapes match themselves when turned, not when folded.
A figure has rotational symmetry if it looks exactly the same after being turned about a fixed point through less than a full turn. The fixed point is the centre of rotation. After a full turn of 360° every figure looks the same, so that turn does not count by itself.
In the figure: A pinwheel with 3 equal blades looks the same after turns of 120° and 240°.
Rule 1: Turn and compare — Turn the figure slowly about its centre. If at some turn smaller than 360° it fits exactly over its old outline, it has rotational symmetry. The smallest such turn tells you how many matching positions it has.
Rotational symmetry and centre of rotation
In the app you can move this figure with your finger.
Open in the app- Easy: A square turned by 90° fits its old outline, so the smallest turn is 90°.
- Medium: A rectangle (not a square) does not fit after 90°, but it does after 180°.
- Exam-type: An equilateral triangle fits after 120°, 240° and 360°.
Rule 2: Rotation without a line of symmetry — A figure can look the same after a turn even if no fold works. A pinwheel with bent blades has no line of symmetry, but it still matches after every quarter turn.
- Easy: The capital letter N has no line of symmetry, yet it looks the same after a half turn (180°).
- Medium: A plus sign (+) looks the same after every 90° turn.
- Exam-type: The centre of rotation of a square is the point where its diagonals meet.
Every figure looks the same after 360°, so a figure has rotational symmetry only if it also matches after a smaller turn.
Thinking every figure has rotational symmetry because of the full turn, or mixing it up with line symmetry.
Rotational symmetry = looks the same after a turn smaller than a full turn. Memory clue: “Turn, Look, Same?”.
Spot the difference
| Figure | Smallest turn that matches | Centre of rotation |
|---|---|---|
| Square | 90° | where the diagonals meet |
| Rectangle | 180° | where the diagonals meet |
| Equilateral triangle | 120° | the middle of the triangle |
| Scalene triangle | none (only 360°) | not needed |
Practice questions
- 1What is the smallest turn after which a square looks the same?
- 2Does a rectangle look the same after a 180° turn?
- 3A pinwheel has 4 equal blades. After how many degrees does it first look the same?
- 4A pinwheel has 6 equal blades. What is the smallest turn after which it looks the same?
- 5Does a triangle with three different sides have rotational symmetry?
- 6Does the capital letter H look the same after a half turn?
- 7A pinwheel first looks the same after a turn of 72°. How many equal blades does it have?
- 8Is a turn of 100° a turn that makes a square look the same? Give a reason.
Show answers (try them yourself first)
- 11) 90°
- 22) Yes
- 33) 90°
- 44) 60° (360 ÷ 6)
- 55) No (only after 360°)
- 66) Yes
- 77) 5 blades (360 ÷ 72)
- 88) No, 100 is not a multiple of 90
Angles of rotational symmetry
A merry-go-round has 6 identical seats. Each time it turns by exactly one seat, the whole ride looks as it did before. The smallest such turn is a fixed part of a full round.
The turns after which a figure looks the same are its angles of symmetry. The smallest one is 360° shared equally among the matching positions, and the others are its repeats, up to 360°. A regular polygon has as many matching positions as it has sides.
In the figure: A triangle matches after 120°, 240° and 360°; a square after 90°, 180°, 270° and 360°.
Rule 1: Find the smallest angle — Divide 360° by the number of matching positions. For a regular polygon that is the number of sides.
Angles of rotational symmetry
In the app you can move this figure with your finger.
Open in the app- Easy: Square: 360 ÷ 4 = 90°.
- Medium: Equilateral triangle: 360 ÷ 3 = 120°.
- Exam-type: Regular hexagon: 360 ÷ 6 = 60°.
Rule 2: List all the angles — Keep adding the smallest angle until you reach 360°. Every number on the way is an angle of symmetry. The last one, 360°, is the full turn.
- Easy: Regular pentagon: 72°, 144°, 216°, 288° and 360°.
- Medium: Rectangle (not a square): 180° and 360°.
- Exam-type: Smallest angle 45°: 360 ÷ 45 = 8 matching positions.
The last angle in the list is always 360°, because a full turn brings every figure back to where it started.
Dividing 180° instead of 360° by the number of sides, or forgetting to list the repeats of the smallest angle.
Smallest angle = 360° ÷ number of matching positions; keep adding it up to 360°. Memory clue: “Three-sixty, shared equally”.
Spot the difference
| Figure | Smallest angle | All angles |
|---|---|---|
| Equilateral triangle | 120° | 120°, 240°, 360° |
| Square | 90° | 90°, 180°, 270°, 360° |
| Regular hexagon | 60° | 60°, 120°, 180°, 240°, 300°, 360° |
| Rectangle | 180° | 180°, 360° |
Practice questions
- 1What is the smallest angle of rotational symmetry of an equilateral triangle?
- 2What is the smallest angle of rotational symmetry of a regular octagon?
- 3Write all the angles of rotational symmetry of a square.
- 4What is the smallest angle of rotational symmetry of a regular polygon with 10 sides?
- 5A figure has smallest angle of symmetry 40°. How many matching positions does it have in a full turn?
- 6What is the second angle of rotational symmetry of a regular pentagon?
- 7The smallest angle of rotational symmetry of a regular polygon is 24°. How many sides does it have?
- 8A figure looks the same after a turn of 120°. Which of 80°, 240° and 300° are also angles of symmetry, if 120° is the smallest?
Show answers (try them yourself first)
- 11) 120°
- 22) 45° (360 ÷ 8)
- 33) 90°, 180°, 270°, 360°
- 44) 36° (360 ÷ 10)
- 55) 9 (360 ÷ 40)
- 66) 144° (2 × 72°)
- 77) 15 sides (360 ÷ 24)
- 88) Only 240° (a multiple of 120°)
Reflection and rotation together
A square window, a triangular warning sign and a slanting tile can all be tested two ways: fold them in a mirror line, and turn them about the centre. Each shape may pass one test, both tests or neither.
Line symmetry is tested by folding and rotational symmetry by turning. A figure can have line symmetry only, rotational symmetry only, both, or neither. The two kinds do not always come together.
In the figure: Square: both. Isosceles triangle: only a line. Parallelogram: only a turn.
Rule 1: Two separate tests — Do the fold test first and count the lines. Then do the turn test and find the smallest turn that matches. Write both results for each figure.
Reflection and rotation together
In the app you can move this figure with your finger.
Open in the app- Easy: Square: 4 lines, and it matches after 90° turns, so it has both kinds.
- Medium: Isosceles triangle: 1 line, but no turn smaller than 360° matches, so only line symmetry.
- Exam-type: Slanting parallelogram: no line, but it matches after 180°, so only rotational symmetry.
Rule 2: Compare the counts — A regular polygon has as many lines as sides and as many matching positions as sides. A rectangle has 2 lines and 2 matching positions. A kite has 1 line and no turn that matches.
- Easy: Equilateral triangle: 3 lines and 3 matching positions (120°, 240°, 360°).
- Medium: Kite: 1 line, no rotational symmetry.
- Exam-type: Bent-blade pinwheel with 4 blades: no line, but matches after every 90°.
Having a line of symmetry does not give rotational symmetry, and having rotational symmetry does not give a line.
Believing that line symmetry and rotational symmetry always come together.
Fold test for lines, turn test for rotation; do both separately. Memory clue: “Fold for lines, Turn for rotation”.
Spot the difference
| Figure | Lines of symmetry | Rotational symmetry |
|---|---|---|
| Square | 4 | yes (90°) |
| Isosceles triangle | 1 | no |
| Parallelogram | 0 | yes (180°) |
| Scalene triangle | 0 | no |
Practice questions
- 1Does a square have both line symmetry and rotational symmetry?
- 2Does an isosceles triangle have rotational symmetry?
- 3Does a slanting parallelogram (not a rectangle) have a line of symmetry?
- 4For a rectangle that is not a square, give the number of lines of symmetry and the smallest matching turn.
- 5Does a kite have rotational symmetry?
- 6For an equilateral triangle, give the number of lines of symmetry and the smallest matching turn.
- 7Which of these has exactly 1 line of symmetry and no rotational symmetry: a square, an isosceles triangle, a parallelogram?
- 8A regular polygon has 6 lines of symmetry and its smallest matching turn is 60°. Name it.
Show answers (try them yourself first)
- 11) Yes
- 22) No
- 33) No
- 44) 2 lines, 180°
- 55) No
- 66) 3 lines, 120°
- 77) The isosceles triangle
- 88) A regular hexagon