L2 · Rotation symmetry

Learning goal

“I can check whether a shape has rotation symmetry by turning it about its centre, I can find the smallest turn that maps it onto itself, and I can state its order of rotation symmetry.”

Rotation symmetry — a shape has it when a turn of less than a full turn (360°), about its centre, makes it look exactly the same.

Centre of rotation — the fixed point a shape turns around; for these shapes, the middle of the shape.

Order of rotation symmetry — the number of times a shape maps onto itself in one full 360° turn, counting the full turn.

Quarter turn — a turn of 90°; a half turn is 180° and a three-quarter turn is 270°.

Worked example

Does a square have rotation symmetry? Mark the centre of the square. Step 1: Turn the square a quarter turn (90°) about its centre. Each side lands where a side was, each vertex lands where a vertex was. The square looks exactly the same. Step 2: So YES - a square has rotation symmetry, and the smallest turn that works is 90°. Step 3: Not every shape passes. Turn a scalene triangle (all sides different) by 90° or 180° about its centre: a long side lands where a short side was. It only looks the same after the full 360° - so it has NO rotation symmetry.

Check yourself — explain it

A shape has rotation symmetry when...

Paired problem — example still visible

Does an equilateral triangle have rotation symmetry? What is the smallest turn? -> Yes. Turn it 120° about its centre: each vertex moves to the next vertex's place, and the triangle looks the same. Smallest turn 120°.

Faded completion — last step is yours

"A regular pentagon turned about its centre first looks the same after a turn of ___°." -> 72° (360 / 5)

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