Symmetry (Years 6–9) — Maths

Fold a paper heart down the middle and both halves match — that is symmetry. Butterflies, snowflakes, letters like A and H, the England flag: symmetry is everywhere, and it simply means a shape looks balanced. There are two kinds: line symmetry (one half mirrors the other) and rotational symmetry (it looks the same as you turn it).

Why this matters: spotting symmetry helps you sketch shapes accurately, check your own drawings, and understand tilings, logos and patterns. It is also the everyday partner of reflection and rotation from the transformations chapter — the mirror lines are the same idea.

A line of symmetry: each half mirrors the other (a square has 4)

Line symmetry

A line of symmetry (mirror line) splits a shape so one side is the exact reflection of the other. The fold test is the best way to picture it: if you could fold the shape along that line, the two halves would land perfectly on top of each other. A shape can have several:

Rectangle → 2 · Square → 4 · Equilateral triangle → 3 · Regular pentagon → 5 · Circle → infinitely many.

Notice the pattern: a regular polygon with n sides has exactly n lines of symmetry. That is your shortcut for pentagons, hexagons, octagons and beyond.

A regular pentagon has 5 lines of symmetry — one through each vertex.

Rotational symmetry

A shape has rotational symmetry if it looks the same after a turn of less than a full circle. The order is how many times it matches itself in one complete turn. Think of a car wheel with identical spokes: as it spins, it snaps into a look-alike position several times per revolution — that count is the order.

A square has rotational symmetry of order 4 — it looks the same every 90° turn (90°, 180°, 270°, 360°). A rectangle has order 2: it matches at 180° and 360° only. Order 1 means no rotational symmetry — it only matches after a full turn.

Every 90° turn about the centre, the square looks the same — order 4.

Shape	Lines of symmetry	Order of rotation
Rectangle	2	2
Square	4	4
Equilateral triangle	3	3
Regular pentagon	5	5
Regular hexagon	6	6
Circle	infinite	infinite

Common mistake: not every line through the middle is a line of symmetry — both halves must mirror exactly. A rectangle's diagonals are not lines of symmetry, even though they pass through the centre, because folding along a diagonal does not make the halves match.

Fold along a rectangle's diagonal and the halves don't match — not a line of symmetry.

In a nutshell: count mirror lines for line symmetry (a regular n-gon has n of them); the order of rotational symmetry is how many times a shape fits onto itself in one full turn.