MathematicsExam code: 0580

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Slicing a solid in two

  • Slice a solid so that the two pieces are identical mirror images, and the flat surface of that cut is a plane of symmetry
  • Two rules cover almost every case asked
    • For a prism, count the mirror lines in the cross-section, then add one more for the cut made halfway along its length
    • For a pyramid, the count is simply the number of mirror lines in the base
  • Applying those rules gives the standard results
    • A cuboid with three different edge lengths has 3
    • A cube has 9: the prism rule gives only 5, but a cube is a prism in all three directions, and counting each plane once over the three gives 9
    • A cylinder has infinitely many
    • A pyramid on a square base has 4
The nine planes of symmetry of a cube shown one at a time as shaded slices, then the three of a cuboid, then the four of a square-based pyramid
Source: Planes of symmetry by Save My Exams
  • The rules run backwards too, which is how the harder version of the question is set

Worked example

Working back from the number of planes

A prism has 7 planes of symmetry, and its cross-section is a regular polygon. Name the polygon.

Solution:

  • For a prism the count is the mirror lines in the cross-section, plus one for the cut halfway along
  • So the cross-section must have 7 − 1 = 6 mirror lines
  • A regular polygon with n sides has exactly n of them, so n = 6
  • The cross-section is a regular hexagon
  • Check: a regular hexagon has 6 lines of symmetry, and 6 + 1 = 7
A square-based pyramid with a vertical axis through its apex, which it matches four times in a full turn because its base is a square, beside a cylinder whose circular cross-section gives it infinitely many matching positions about the same kind of axis
Source: Planes of symmetry by Save My Exams
  • A solid can also have rotational symmetry about an axis, and the syllabus asks you to recognise this for prisms, cylinders, pyramids and cones

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