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".
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.
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.
Rectangle diagonals are not mirror lines
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.
Authored from a well-established symmetry pitfall; the Nov 2024 Higher reports were read and held no symmetry question.
Lines of symmetry ≠ rotational order
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 .
Authored from a well-established symmetry pitfall; not flagged in the Nov 2024 Higher reports, which contained no symmetry question.
Rotational order is never 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.
Authored from a well-established symmetry pitfall; the Nov 2024 Higher examiner reports (the exam and 2H) were checked and did not feature a symmetry question.
Miscounting lines of symmetry
Students double-count the same line, or miss the diagonal ones. A regular hexagon has lines (through opposite vertices AND opposite edge-midpoints); a regular octagon has , not .
Cube 9 planes, cuboid 3
A cube has planes of symmetry, but a cuboid with three different edge lengths has only (one parallel to each pair of faces). Do not award a plain cuboid the cube's planes.
Sign error in x = −b/2a
For the quadratic axis , students forget the leading minus or the denominator. With , : , not or .
Trace, fold and rotate to check
For line symmetry, imagine folding along each candidate line. For rotational symmetry, trace the shape and turn the tracing about the centre, counting matches in one full turn. Mark the centre and mirror lines so you don't double-count.
Axis of symmetry finds the vertex
For a parabola, find the axis (or the midpoint of the two roots), then substitute back to get the turning-point . The vertex always lies on the axis of symmetry.
Order and lines are separate counts
Count lines of symmetry and rotational order SEPARATELY — they match only for regular polygons. State each with its own working; do not assume one equals the other.
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 a line of symmetry?
A regular octagon is drawn. Write down the number of lines of symmetry it has.
