Symmetry

Maths · Geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A flat shape has line symmetry if a line can fold it into two halves that fit exactly on each other. Some shapes have several such lines and some have none.

The order of rotational symmetry is the number of times a shape looks exactly the same during one full turn. Every shape fits onto itself after a full turn, so a shape with no rotational symmetry has order 1, not 0.

Solids have symmetry too. A plane of symmetry cuts a solid into two halves that are mirror images. A solid may also have rotational symmetry about an axis, such as the line through the top of a pyramid and the centre of its base.

Rules to remember

Regular polygon with n sides: n lines of symmetry and rotational symmetry of order n
Square: 4 lines, order 4. Rectangle: 2 lines, order 2. Rhombus: 2 lines (its diagonals), order 2
Parallelogram: 0 lines, order 2. Kite: 1 line, order 1. Isosceles trapezium: 1 line, order 1
Equilateral triangle: 3 lines, order 3. Isosceles triangle: 1 line, order 1. Scalene triangle: 0 lines, order 1
Cuboid with three different edge lengths: 3 planes of symmetry. Cube: 9 planes of symmetry
Right pyramid on a regular n-sided base: n planes of symmetry (a regular tetrahedron has 6) and rotational symmetry of order n about its axis
Cylinder and cone: an infinite number of planes of symmetry through the axis (a cylinder has one more, halfway along its length)

Common mistake

Students say that a parallelogram has two lines of symmetry along its diagonals. Folding along a diagonal does not make the halves fit, so it has no lines of symmetry, only rotational symmetry of order 2.

Worked example

A prism has a cross-section that is an equilateral triangle. How many planes of symmetry does the prism have?

An equilateral triangle has 3 lines of symmetry.
Each line gives one plane that runs along the length of the prism: 3 planes.
One more plane cuts the prism halfway along its length, parallel to the two ends: 1 plane.
Total: 3 + 1 = 4.

Answer: 4 planes of symmetry

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