Line (reflective) symmetry
A line of symmetry is a mirror line: fold the shape along it and the two halves match exactly. Count every such line. A regular -sided polygon has exactly lines of symmetry; a circle has infinitely many.
Rotational symmetry and its ORDER
The order of rotational symmetry is how many times a shape looks identical to its start during one full turn. A regular -gon has order . Every shape maps onto itself after a full turn, so the order is never less than — order means "no rotational symmetry".
Plane symmetry of 3D solids
A plane of symmetry slices a solid into two mirror-image halves. A cuboid with three different edge lengths has planes of symmetry; a cube has ; a prism has the planes of its cross-section plus the one cutting it in half along its length; a cylinder has infinitely many (every plane through its axis, plus one horizontal one).
Symmetry of a quadratic graph
A parabola is symmetric about the vertical line , which passes through its turning point and the midpoint of its two roots.
Drawn from real examiner reports.
Counting a full turn as rotational symmetry order 0
Some students give the order as for a shape with no rotational symmetry (e.g. a scalene triangle). The minimum order is , because every shape fits onto itself after a complete rotation.
Confusing lines of symmetry with order of rotational symmetry
Students assume the two counts are always equal. They match for regular polygons, but not in general: a parallelogram has lines of symmetry yet rotational order ; an isosceles triangle has line but rotational order .
Diagonals of a rectangle are NOT lines of symmetry
A non-square rectangle has only lines of symmetry (through the midpoints of opposite sides) — its diagonals are not mirror lines, because folding along a diagonal does not map the rectangle onto itself.
Trace, fold and rotate to check
For line symmetry, imagine folding along each candidate line. For rotational symmetry, picture tracing the shape and turning the tracing about the centre — count the matches in one full turn. Mark the centre and any mirror lines on the diagram so you don't miss or double-count one.
A line of symmetry is a mirror line: fold the shape along it and the two halves land exactly on top of each other.
| Shape | Lines of symmetry |
|---|---|
| Isosceles triangle | |
| Equilateral triangle | |
| Rectangle (not square) | |
| Rhombus | (the diagonals) |
| Square | |
| Regular pentagon | |
| Regular -gon | |
| Parallelogram (not a rhombus/rectangle) | |
| Circle | infinitely many |
Watch out: a rectangle's diagonals are not lines of symmetry — only the two lines through the midpoints of opposite sides are.
What is the order of rotational symmetry?
Write down the order of rotational symmetry of a regular hexagon.
