Practice Maths

Rotational Symmetry

Key Terms

Rotational symmetry
A shape has rotational symmetry if it looks exactly the same after being rotated by less than 360° about its centre.
Order of rotational symmetry
The number of times a shape looks identical in one full 360° rotation (including at 360°). Order 1 means no rotational symmetry.
Minimum angle of rotation
The smallest angle needed to rotate the shape onto itself: 360° ÷ order.
Centre of rotation
The fixed point around which the shape is rotated — usually the centre of the shape.
Square Order 4 — every 90° Equil. Triangle Order 3 — every 120° Rectangle Order 2 — every 180° Parallelogram Order 2 — 0 lines of symmetry!

Red dots mark the centre of rotation. Each shape maps onto itself at the angle shown.

Orders for Common Shapes

  • Equilateral triangle: order 3, angle 120°
  • Square: order 4, angle 90°
  • Rectangle: order 2, angle 180°
  • Regular pentagon: order 5, angle 72°
  • Regular hexagon: order 6, angle 60°
  • Parallelogram: order 2, angle 180°
  • Rhombus: order 2, angle 180°
  • Isosceles triangle: order 1 (no rotational symmetry)
  • Scalene triangle: order 1 (no rotational symmetry)
  • Circle: infinite order
  • Regular n-gon: order n, angle 360° ÷ n
Hot Tip
Don’t confuse the order of rotational symmetry with the number of lines of symmetry. A rectangle has 2 lines of symmetry AND order 2 rotational symmetry. A rhombus also has 2 lines AND order 2 rotational symmetry. But a parallelogram has 0 lines of symmetry and yet order 2 rotational symmetry!

Worked Example 1 — Square

Find the order and minimum angle of rotational symmetry for a square.

A square looks the same after rotating 90°, 180°, 270°, and 360°.

Order = 4. Minimum angle = 360° ÷ 4 = 90°.

Worked Example 2 — Equilateral Triangle

Find the order and minimum angle of rotational symmetry for an equilateral triangle.

An equilateral triangle looks the same at 120°, 240°, and 360°.

Order = 3. Minimum angle = 360° ÷ 3 = 120°.

Square — order 4 angle = 90° Equil. Δ — order 3 angle = 120° Rectangle — order 2 angle = 180° Hexagon — order 6

Red dots show the centre of rotation. Red dashes show axes of symmetry.

What Is Rotational Symmetry?

A shape has rotational symmetry if it looks exactly the same as the original after being rotated by an angle less than 360° around its centre. The shape "maps onto itself" during the rotation.

For example, a square looks the same after a 90° rotation, a 180° rotation, or a 270° rotation — before you complete the full 360°. So a square has rotational symmetry.

Every shape technically looks the same after a full 360° rotation, so we don't count that — rotational symmetry only counts rotations less than 360°.

Order of Rotational Symmetry

The order of rotational symmetry is the number of times a shape looks identical to the original during one full 360° rotation (including the final 360° position). If a shape has no rotational symmetry, its order is 1.

  • Equilateral triangle: looks the same at 120°, 240°, 360° → order 3
  • Square: looks the same at 90°, 180°, 270°, 360° → order 4
  • Regular hexagon: looks the same at 60°, 120°, 180°, 240°, 300°, 360° → order 6
  • Rectangle (not square): looks the same at 180° and 360° → order 2
  • Scalene triangle: only looks the same at 360° → order 1 (no rotational symmetry)

Formula for Regular Polygons

For a regular polygon with n sides, the order of rotational symmetry is n, and the shape maps onto itself every 360° ÷ n.

  • Regular pentagon (5 sides): order 5, rotates every 72°
  • Regular octagon (8 sides): order 8, rotates every 45°

Rotational Symmetry vs Line Symmetry

A shape can have one type of symmetry without the other. An equilateral triangle has both 3 lines of symmetry AND rotational symmetry of order 3. A regular star can have rotational symmetry but fewer lines of symmetry. A scalene triangle has neither. These are different properties — don't confuse them.

Key tip: To find the order of rotational symmetry, ask: "how many times does this shape look the same as I rotate it through 360°?" An easy way to check is to mark one vertex with a dot, then rotate the shape step by step. Each time the shape looks the same (dot position aside), that counts as one match. The total count (including when you return to 360°) is the order.

Mastery Practice

  1. Order of Rotational Symmetry — Shapes Fluency

    State the order of rotational symmetry and the minimum angle of rotation for each shape. The red dot marks the centre of rotation. Write “order 1 (no rotational symmetry)” where appropriate.

    (a) Equilateral Δ
    (b) Square
    (c) Rectangle
    (d) Regular hexagon
    (e) Parallelogram
    (f) Rhombus
    (g) Isosceles Δ
    (h) Circle
    1. Equilateral triangle
    2. Square
    3. Rectangle
    4. Regular hexagon
    5. Parallelogram
    6. Rhombus
    7. Isosceles triangle
    8. Circle

  2. True or False Fluency

    Write True or False for each statement.

    1. A shape with order of rotational symmetry 1 has NO rotational symmetry.
    2. A circle has infinite rotational symmetry.
    3. A parallelogram has 2 lines of symmetry.
    4. A rhombus has order 2 rotational symmetry.
    5. The minimum angle of rotation for a regular hexagon is 60°.
    6. A square has order 4 rotational symmetry.
    7. Every shape has at least order 1 rotational symmetry.
    8. A rectangle has rotational symmetry of order 2.

  3. Find the Minimum Angle Fluency

    Calculate the minimum angle of rotation for a shape with the given order of rotational symmetry.

    1. Order 2
    2. Order 3
    3. Order 4
    4. Order 5
    5. Order 6
    6. Order 8
    7. Order 9
    8. Order 12

  4. Compare Line and Rotational Symmetry Understanding

    For each shape, state: (i) the number of lines of symmetry and (ii) the order of rotational symmetry.

    1. Square
    2. Rectangle
    3. Rhombus
    4. Parallelogram
    5. Equilateral triangle
    6. Regular hexagon

  5. Identifying Shapes from Properties Understanding

    1. A shape has 4 lines of symmetry and rotational symmetry of order 4. Name the shape.
    2. A shape has 0 lines of symmetry and rotational symmetry of order 2. Name the shape.
    3. A shape has 1 line of symmetry and NO rotational symmetry. Give an example.
    4. A shape has 5 lines of symmetry and order 5 rotational symmetry. Name the shape.
    5. A shape has infinite lines of symmetry and infinite order of rotation. Name the shape.

  6. Real World Rotational Symmetry Problem Solving

    1. A car tyre has 5 identical bolts equally spaced. What is the order of rotational symmetry of the bolt pattern? What is the minimum rotation angle?
    2. A starfish has 5 arms equally spaced. What is its order of rotational symmetry?
    3. A ceiling fan has 4 identical blades equally spaced. What is the minimum angle needed to rotate it so it looks the same?
    4. Design (describe in words) a shape or pattern that has order 6 rotational symmetry. Where is the centre of rotation?
    5. A clock face (without numbers) has 12 identical hour marks. What is the order of rotational symmetry? What is the minimum rotation angle?
    6. The recycling symbol has 3-fold rotational symmetry. What is its minimum angle of rotation? Does it also have line symmetry?

  7. Find the Order from the Angle Fluency

    A shape looks the same after the given rotation. Find the order of rotational symmetry.

    1. Minimum rotation = 180°
    2. Minimum rotation = 120°
    3. Minimum rotation = 90°
    4. Minimum rotation = 72°
    5. Minimum rotation = 60°
    6. Minimum rotation = 45°
    7. Minimum rotation = 40°
    8. Minimum rotation = 30°

  8. Rotational Symmetry — Deeper Thinking Understanding

    1. A shape has order 4 rotational symmetry. List all the angles of rotation (less than 360°) for which it looks the same.
    2. A shape has order 6 rotational symmetry. List all the angles of rotation (less than 360°) for which it looks the same.
    3. A shape has rotational symmetry of order n. How many angles of rotation (less than 360°) make it look identical? Write a general rule.
    4. Can a shape have rotational symmetry but no line symmetry? Give an example.
    5. Can a shape have line symmetry but no rotational symmetry? Give an example.

  9. Symmetry of Composite Designs Understanding

    (a) Divided hexagon
    (b) Square with corner arrows
    (c) Star of David
    4-blade 3-blade
    (d) Compare fan blades
    1. A regular hexagon is divided into 6 equilateral triangles by drawing lines from each vertex to the centre. The design has the same rotational symmetry as the hexagon. What is the order and minimum angle?
    2. A square has its four corners marked with four identical arrows, each pointing outward from the centre. The design looks the same after a 90° rotation. What is the order of symmetry?
    3. Two identical equilateral triangles are placed point-to-point (forming a Star of David pattern — a 6-pointed star). What is the order of rotational symmetry of the 6-pointed star?
    4. A 4-bladed fan and a 3-bladed fan: which has a smaller minimum rotation angle? What is each angle?

  10. Rotational Symmetry in Design and Nature Problem Solving

    (a) Snowflake design
    (b) Gear wheel
    (c) Regular pentagon
    1. A snowflake has 6-fold rotational symmetry. A designer wants to create a pattern based on a snowflake. What is the minimum rotation angle? If the designer rotates the pattern 240°, does it look the same? Justify your answer.
    2. A gear wheel has 15 identical teeth equally spaced around its circumference. What is the order of rotational symmetry? What is the minimum rotation angle?
    3. A football (soccer ball) viewed from the top shows a regular pentagon in the centre. The visible pattern has 5-fold rotational symmetry. Through how many degrees must you rotate the ball for it to look the same?
    4. Create your own example: describe a real-world object or logo not already mentioned in class that has rotational symmetry. State its order and minimum rotation angle. Justify your claim.
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