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 looks identical after being rotated by some angle less than 360°360° about a central point. The order of rotational symmetry is the number of distinct positions where the figure looks the same during a full rotation.

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 figure has rotational symmetry if turning it less than a full circle about a center makes it look unchanged.

Common stuck point: The procedure for rotational symmetry is the easy part; the trap is counting the full 360°360° position as extra order. Asking "Does the figure coincide with itself after a rotation of less than 360°360° about a center point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the figure coincide with itself after a rotation of less than 360°360° about a center point?

Worked Examples

Example 1

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

Answer

Order 66; minimum rotation angle 60°60°.

First step

1
A regular hexagon has 66 equal sides and 66 equal angles.

Full solution

  1. 2
    Minimum angle: 360°6=60°\dfrac{360°}{6} = 60°.
  2. 3
    It maps onto itself at rotations of 60°,120°,180°,240°,300°,360°60°, 120°, 180°, 240°, 300°, 360° — that is 66 positions.
  3. 4
    The order of rotational symmetry is 66.
A regular nn-gon has rotational symmetry of order nn, with minimum angle 360°n\frac{360°}{n}. Each additional symmetry position is a multiple of this smallest angle up to 360°360°.

Example 2

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

Example 3

medium
Does a rectangle (non-square) have rotational symmetry? Give the order and smallest angle.

Example 4

medium
Does the letter 'N' have rotational symmetry? Explain.

Example 5

medium
Does a regular star (5-pointed star with equal points) have rotational symmetry? State order and angle.

Example 6

hard
A figure has rotational symmetry such that rotations of 40°40°, 80°80°, 120°120°, ... all map it onto itself. What is its order?

Example 7

hard
A snowflake has rotational symmetry of order 66. List all angles of rotation that map it to itself between 0°0° (exclusive) and 360°360° (inclusive).

Example 8

hard
A regular nn-gon has rotational symmetry of order nn. Which nn gives a smallest rotation angle of 30°30°?

Example 9

challenge
Prove that the set of rotational symmetries of a regular nn-gon forms a cyclic group of order nn.

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 44. List all angles of rotation that map it onto itself.

Example 2

medium
Equilateral triangle ABCABC is centred at the origin. A 120°120° counterclockwise rotation maps A→BA \to B. What does CC map to, and what is the order of symmetry?

Example 3

easy
A figure has rotational symmetry of order 66. List all angles less than 360°360° that map it onto itself.

Example 4

medium
Does a scalene triangle have rotational symmetry?

Example 5

medium
A regular polygon has smallest rotation angle 24°24°. How many sides does it have?

Example 6

medium
A figure has rotational symmetry of order 88. What is the smallest rotation angle that maps it onto itself?

Example 7

hard
Does a regular hexagon have rotational symmetry of order 33? Explain.

Example 8

hard
A pinwheel has 44 identical blades arranged around a center. What is its rotational symmetry order and smallest angle?

Example 9

medium
What is the rotational symmetry order of a non-square rhombus?

Example 10

medium
How many angles strictly between 0°0° and 360°360° map a figure of rotational order 77 onto itself?
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Background Knowledge

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

symmetryrotationangle relationships