Line and rotational symmetry

Symmetry maps the whole figure to itself

A line of symmetry gives matching mirror halves. A square has four such lines; a non-square rectangle has two. The order of rotational symmetry counts the matching positions in one full turn, including the full turn: a square has order 4 and a non-square rectangle order 2. A figure with only the full-turn match has order 1.

Equal area alone does not establish symmetry. Test where every corner and edge would go under the proposed reflection or rotation.

Test a shape rather than its appearance

A non-square rectangle matches itself after a half turn and a full turn, giving rotational order 2. A square also matches after quarter and three-quarter turns, giving order 4. A general parallelogram has rotational order 2 but usually no line of symmetry; rotation and reflection are different tests.

Check your understanding 1 A full turn still counts

A plane figure matches itself at turns of 120°, 240° and 360°, with no smaller positive matching turn. Give its rotational order. Does that determine its number of symmetry lines?

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Its rotational order is 3. The rotation data alone do not determine reflection symmetry; a figure can have threefold rotation without any mirror line.

State the centre and the proposed line

For rotation, turn around the stated centre rather than sliding the figure. For reflection, every point must match at equal perpendicular distance across the proposed line. Equal areas on two sides are not sufficient.