01 — Explanation
What is line symmetry?
A shape has line symmetry if a mirror line can be drawn across it so that one half is the exact reflection of the other — every point on one side has a matching point on the opposite side, the same distance from the mirror line. A shape can have more than one line of symmetry; a circle, for instance, has infinitely many, since any diameter acts as a mirror line.
02 — Explanation
What is rotational symmetry, and how do I find its order?
A shape has rotational symmetry if turning it about its centre, through some angle less than a full turn, makes it land back on itself exactly. The order of rotational symmetry counts how many times this happens during one complete turn — equivalently, it's the number of identical positions the shape passes through as it spins once around.
Common mistake
writing "order 0" for a shape with no rotational symmetry. Every shape matches its start position after a full turn, so the lowest possible order is 1, never 0 — a shape with no extra rotational symmetry beyond the trivial full turn is simply described as having order 1.
03 — Explanation
What are the symmetry properties of common shapes?
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Isosceles triangle (not equilateral) | 1 | 1 |
| Scalene triangle | 0 | 1 |
| Square | 4 | 4 |
| Rectangle (not a square) | 2 | 2 |
| Rhombus (not a square) | 2 | 2 |
| Parallelogram (not a rhombus) | 0 | 2 |
| Kite | 1 | 1 |
| Regular polygon with sides |
A rhombus and a square both have 4 equal sides, but only the square has 4 equal angles too — that extra condition is what gives it 4 lines of symmetry instead of 2. Symmetry depends on the whole shape, not side lengths alone. The rhombus's lines of symmetry run along its diagonals, whereas the square's four lines of symmetry are its two diagonals and the two lines joining the midpoints of opposite sides — the square's extra right-angle condition unlocks two additional mirror lines that the rhombus simply doesn't have.
04 — Explanation
How do I find the symmetry of a shape built from two simpler shapes?
Combining shapes can create or destroy symmetry compared to the original pieces, so always look at the finished combined outline rather than assuming the symmetry of each piece carries over automatically.
Worked example. A rectangle has a semicircle attached to one of its shorter ends, with the semicircle's diameter equal to that end's length. State the number of lines of symmetry of the resulting shape.
The shape is symmetric about the horizontal line running through its centre lengthwise (the semicircle's straight edge sits on the rectangle, and both halves above and below that centre line mirror each other), but it is not symmetric left-to-right, since one end is flat and the other is curved. This composite shape therefore has exactly 1 line of symmetry, and no rotational symmetry beyond order 1, since rotating it would not map the flat end onto the curved end.
Worked example. A regular pentagon is drawn. State (a) its number of lines of symmetry, and (b) its order of rotational symmetry.
A regular polygon's lines of symmetry and order of rotational symmetry both equal its number of sides: a pentagon has 5 sides, so it has 5 lines of symmetry and order 5 rotational symmetry.