Rotational Symmetry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rotational Symmetry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A figure has rotational symmetry if it matches itself after a rotation less than 360^circ.

If you turn it and it still fits exactly, it has rotational symmetry.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A shape has rotational symmetry if it looks identical after being rotated by some angle less than 360°.

Common stuck point: Students count full-turn matches that do not indicate nontrivial symmetry.

Sense of Study hint: Test the smallest angle that maps the figure onto itself.

Worked Examples

Example 1

easy
Determine the order of rotational symmetry and the minimum angle of rotation for a regular hexagon.

Solution

  1. 1
    A regular hexagon has 6 equal sides and 6 equal angles.
  2. 2
    Minimum angle: \dfrac{360°}{6} = 60°.
  3. 3
    It maps onto itself at rotations of 60°, 120°, 180°, 240°, 300°, 360° — that is 6 positions.
  4. 4
    The order of rotational symmetry is 6.

Answer

Order 6; minimum rotation angle 60°.
A regular n-gon has rotational symmetry of order n, with minimum angle \frac{360°}{n}. Each additional symmetry position is a multiple of this smallest angle up to 360°.

Example 2

medium
Which of the letters 'S' and 'H' has rotational symmetry? State the order and angle for each that qualifies.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A figure has rotational symmetry of order 4. List all angles of rotation that map it onto itself.

Example 2

medium
Equilateral triangle ABC is centred at the origin. A 120° counterclockwise rotation maps A \to B. What does C map to, and what is the order of symmetry?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

symmetryrotationangle relationships