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Rotational Symmetry

FoundationChallengeAQAEdexcelOCR

Practise rotational symmetry with this free Foundation GCSE Maths worksheet from Virtus Academy. You'll work through finding the order of rotational symmetry, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. The order is how many times a shape looks the same in a full turn.

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Topic overview

A shape has rotational symmetry if it looks the same after being turned by less than a full circle. The number of positions in which it matches is called the order of rotational symmetry.

A square has order \(4\), because it looks identical after quarter turns. An equilateral triangle has order \(3\), and a rectangle has order \(2\).

Every shape looks the same after a full \(360^\circ\) turn, so the minimum order is \(1\). A shape with order \(1\) is usually described as having no rotational symmetry, since the only match is the complete turn.

Revision notes

Finding the order

Count how many times the shape matches its original position during one full turn.

For a regular polygon, the order equals the number of sides, so a regular hexagon has order \(6\).

Order 1

A shape that only matches after a full turn has order \(1\), described as having no rotational symmetry.

An ordinary scalene triangle is an example.

Rotational versus line symmetry

They are independent properties. A parallelogram has rotational symmetry of order \(2\) but no lines of symmetry.

A regular polygon has both, with the order and the number of lines both equal to the number of sides.

Key points

  • Rotational symmetry means matching during a turn.
  • The order is how many times it matches in a full turn.
  • A square has order 4.
  • A regular polygon's order equals its number of sides.
  • Order 1 means no rotational symmetry.
  • Line and rotational symmetry are independent.

Worked examples

Example 1

What is the order of rotational symmetry of a square?

Working

\[\text{It matches after each quarter turn}\]count the matching positions
\[4\]state the order

Example 2

What is the order of rotational symmetry of a regular pentagon?

Working

\[\text{A regular polygon's order equals its sides}\]apply the rule
\[5\]state the order

Example 3

A parallelogram has how many lines of symmetry and what order of rotational symmetry?

Working

\[0 \text{ lines of symmetry}\]no fold produces matching halves
\[\text{Order } 2\]it matches after a half turn

Common mistakes

  • Giving order 0.

    The minimum is 1, since every shape matches after a full turn.

  • Confusing rotational order with lines of symmetry.

    They are different properties and can differ, as with a parallelogram.

  • Counting the starting position twice.

    Count matches during one complete turn, including the return to the start, once.

  • Assuming a shape with no lines of symmetry has no rotational symmetry.

    A parallelogram has order 2 despite having no mirror lines.

Exam tips

  • Trace the shape and turn it to count the matching positions.
  • Remember a regular polygon's order equals its number of sides.
  • Keep rotational and line symmetry separate in your mind.
  • The minimum order is 1, never 0.

Key terms

Rotational symmetry
Looking the same after a turn of less than a full circle.
Order
The number of matching positions in a full turn.
Regular polygon
A polygon with equal sides and angles.
Scalene
A triangle with no equal sides.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.