E4.5 Symmetry

Learning objectives

E4.5.1Line and rotational symmetry• Recognise line symmetry and order of rotational symmetry in two dimensions, including related properties of triangles, quadrilaterals and polygons.E4.5.2Symmetry of solids• Recognise planes and axes of symmetry of prisms, cylinders, pyramids and cones.

A line of symmetry divides a flat shape into mirror-image halves. Rotational symmetry describes how many times a shape matches itself during one complete $360^\circ$ turn.

For line symmetry, imagine folding along a candidate line: every point must meet a matching point the same perpendicular distance on the other side. For rotational symmetry, keep the centre fixed and count the matching positions in a full turn, including the final $360^\circ$ position.

Shape	Lines of symmetry	Rotational order
equilateral triangle	3	3
non-equilateral isosceles triangle	1	1
scalene triangle	0	1
square	4	4
non-square rectangle	2	2
non-square rhombus	2	2
general parallelogram	0	2
general kite	1	1
regular $n$ -gon	$n$	$n$

A regular decagon matches after every $36^\circ$ , so its rotational order is $360\div36=10$ ; it also has 10 lines of symmetry. A rhombus has its two diagonals as mirror lines and rotational order 2.

Every shape has rotational order at least 1 because it matches after a full turn. Do not assume a diagonal is a mirror line: in a general rectangle the diagonals are not lines of symmetry, while in a rhombus they are.

In three dimensions, a plane of symmetry cuts a solid into mirror-image halves, while an axis of rotational symmetry is a line about which the solid can rotate and match itself before a full turn.

Solid	How to locate symmetry
right prism	extend each symmetry line of its cross-section along the prism; its lengthwise rotational axis inherits the cross-section's rotational order
right cylinder	any plane through the central axis is a mirror plane, as is the mid-plane parallel to the circular ends; the central axis is rotational
regular pyramid	mirror planes pass through the apex, central axis and symmetry lines of the base; the apex-to-base-centre line is the rotational axis
right circular cone	planes through the apex and central axis are mirror planes; the central axis is rotational

Test a proposed plane by reflecting the whole solid—faces, edges and vertices must coincide. Test a proposed axis by rotating the whole solid around that line; matching only the base is not enough unless the rest of the solid also maps onto itself.

A regular square-based pyramid has four vertical planes of symmetry, each passing through the apex and a symmetry line of the square base. Its rotational axis joins the apex to the centre of the base and has order 4.

A plane is a two-dimensional slice, not a line drawn on one visible face. The exact number of symmetries of a prism or pyramid depends on its cross-section or base; do not transfer the symmetry count from a special regular example to every solid of that family.

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