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Geometry Toolkit

Geometry

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
    • 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
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
Exam tip

Give the precise mathematical name and nothing else — "rhombus" not "diamond", "trapezium" not "trapezoid", "cuboid" not "box". Where a shape is defined by properties, check every property before answering, because several quadrilaterals share any single one; the number of lines of symmetry is usually what separates a square from a rhombus. Diagonal lines of symmetry are the ones most often missed.

Worked example

Working back from the number of planes

A prism has 6 planes of symmetry and its cross-section is a regular polygon. Name that 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 6 − 1 = 5 mirror lines
  • A regular polygon with n sides has exactly n of them, so n = 5
  • The cross-section is a regular pentagon
  • Check: a regular pentagon has 5 lines of symmetry, and 5 + 1 = 6
  • A solid can also have rotational symmetry about an axis, and the syllabus asks you to recognise this for prisms, cylinders, pyramids and cones