Symmetry

A figure has line symmetry if there is a line such that reflecting the figure across it gives the same figure back.

How many lines? Common cases.

• Isosceles triangle: 1 (the perpendicular bisector of the base).
• Equilateral triangle: 3 (each through a vertex and the midpoint of the opposite side).
• Square: 4 (two diagonals + two axes through edge midpoints).
• Rectangle (not square): 2 (two axes through edge midpoints).
• Rhombus (not square): 2 (the two diagonals).
• Regular pentagon: 5. Regular hexagon: 6. Regular $n$-gon: $n$.
• Circle: infinitely many (every diameter is a line of symmetry).
• Parallelogram (not rhombus or rectangle): 0.

Tip. When asked to find the lines of symmetry on a custom shape, fold (mentally or actually) the shape along candidate lines. If the two halves coincide, it's a line of symmetry.

A figure has rotational symmetry of order $n$ if rotating it by $\dfrac{360\degree}{n}$ about its centre returns it to its original position.

Memorise.

• Equilateral triangle: order 3 (rotations of $120\degree$).
• Square: order 4 (rotations of $90\degree$).
• Regular $n$-gon: order $n$ (rotations of $\dfrac{360}{n}\degree$).
• Rectangle (not square): order 2 (rotations of $180\degree$).
• Parallelogram (general): order 2.
• Circle: infinite order.

Trivial order. Every figure has order at least 1 (rotating by $360\degree$ always returns it to itself). Cambridge expects "order 1" → "no rotational symmetry".

Worked. A regular hexagon has rotational symmetry of order $6$. The rotations are by $60\degree, 120\degree, 180\degree, 240\degree, 300\degree, 360\degree$.

Plane of symmetry. A 2D plane that cuts the solid into two mirror-image halves.

Worked. A cuboid has 3 planes of symmetry (one parallel to each pair of opposite faces, each cutting through the centre). A cube has 9 (3 face-parallel + 6 diagonal).

Axis of rotational symmetry. A line about which the solid can be rotated to match itself.

Worked. A cylinder has rotational symmetry of infinite order about the axis through its centre.

Memorise.

• Cuboid (general, three different edge lengths): 3 planes of symmetry, three axes of order 2.
• Cube: 9 planes of symmetry, multiple rotational axes.
• Sphere: infinite planes (any plane through the centre), infinite axes.
• Cylinder (right circular): one axis of infinite-order rotation; infinite planes through that axis.

Cambridge tip. When asked to count planes of symmetry on a solid, sketch the solid and look for natural cuts: through opposite faces, through opposite edges, etc.