Every symmetry question from 11 Cambridge IGCSE Mathematics (0580) past papers (May/June 2025 to May/June 2026), syllabus section 4.5. 11 questions covering Core and Extended tiers, ordered from Core to Extended. Each question is labelled with its question paper code (for example 0580/31/M/J/26) and every final answer matches the official Cambridge mark scheme. Try each question first, then click “Show solution” to check your working. This page is part of our IGCSE Maths 0580 Geometry past paper questions.
0580 Symmetry past paper questions: Core (Papers 1 and 3)
May/June 2026 · Paper 3 · 0580/31/M/J/26 · Q9
(a) (i) On the diagram, draw the line of symmetry. (ii) On the diagram, shade one more square so that it has one line of symmetry.
(b) On the diagram, shade one more square so that it has rotational symmetry of order 2.
Show solution
- Try folding the pattern along each possible line in your head. Only one fold makes the shaded squares match.
- Draw the diagonal from the top-right corner to the bottom-left corner across the full grid.
- The four corner squares already match in every direction. The one extra square in row 2, column 2 decides where the line can be.
- To make a vertical line of symmetry, shade its mirror image in row 2, column 3. For a horizontal line, shade row 3, column 2.
- Shade row 2, column 3 (vertical line of symmetry) or row 3, column 2 (horizontal line of symmetry)
- Order 2 means the pattern looks the same after a half-turn (180°) about the centre.
- A half-turn sends the square in row 2, column 2 to row 3, column 3, so that square must be shaded too.
- Shade row 3, column 3
May/June 2026 · Paper 3 · 0580/33/M/J/26 · Q5
(a) Shade some triangles so that the shape has rotational symmetry of order 3.
(b) Shade one small triangle so that the shape has two lines of symmetry.
Show solution
- Rotational symmetry of order 3 means the shape looks the same after turning one third of a full turn (120°), three times.
- Shade triangles so that each shaded triangle has a matching shaded triangle 120° and 240° round the centre.
- Look for the two mirror lines that the existing pattern almost has.
- Shade the one triangle that makes both halves match across each mirror line.
Feb/Mar 2026 · Paper 3 · 0580/32/F/M/26 · Q3
Draw all the lines of symmetry on this shape.
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- The shape is a regular pentagon inside a circle, and a regular pentagon has as many lines of symmetry as it has sides.
- Each line goes from a corner of the pentagon through the centre to the middle of the opposite side.
- The shape has 5 lines of symmetry. Rule every line right across the shape.
Oct/Nov 2025 · Paper 1 · 0580/11/O/N/25 · Q6
Shade one square so that the diagram has rotational symmetry of order 4.
Show solution
- Rotational symmetry of order 4 means the pattern looks the same after each quarter-turn (90°).
- Turn the pattern a quarter-turn at a time and compare: three corners each have a matching set of shaded squares, but one corner is missing a square.
- Shade the square in row 6, column 7, so all four corners match.
Oct/Nov 2025 · Paper 1 · 0580/12/O/N/25 · Q5
The diagram is made from equilateral triangles. (a) On the diagram, draw all the lines of symmetry. (b) Write down the order of rotational symmetry.
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- A line of symmetry folds the shape so both halves match exactly.
- This shape is built like an equilateral triangle, so it has a line of symmetry from each corner to the middle of the opposite side.
- The shape has 3 lines of symmetry. Draw each one across the whole shape.
- Turn the shape round its centre. It fits onto itself 3 times in a full turn (every 120°).
- Rotational symmetry of order 3
May/June 2025 · Paper 1 · 0580/12/M/J/25 · Q13
(a) A quadrilateral has the geometrical properties: 4 equal length sides, 2 lines of symmetry, rotational symmetry of order 2. Write down the mathematical name of this quadrilateral.
(b) Write down two geometrical properties of a rectangle.
(c) The parallel sides of a trapezium have lengths 6 cm and 4 cm. The area of the trapezium is 15 cm². On the 1 cm² grid, draw a trapezium with these lengths and area.
Show solution
- Four equal sides means a square or a rhombus. A square has 4 lines of symmetry, but this shape has only 2.
- The shape is a rhombus
- Write any two correct facts that are true for every rectangle, for example:
- all angles are 90°; opposite sides are equal; opposite sides are parallel; diagonals are equal and bisect each other; 2 lines of symmetry; rotational symmetry of order 2
- Area of a trapezium = 1⁄2(a + b)h, where a and b are the parallel sides and h is the height.
- Substitute: 1⁄2 × (6 + 4) × h = 15.
- Simplify: 5h = 15, so h = 3 cm.
- Draw a trapezium with parallel sides 6 cm and 4 cm that are 3 cm apart.
0580 Symmetry past paper questions: Extended (Papers 2 and 4)
Feb/Mar 2026 · Paper 2 · 0580/22/F/M/26 · Q2
The diagram shows a parallelogram. Complete the statement: a parallelogram has … lines of symmetry and rotational symmetry of order …
Show solution
- A line of symmetry folds the shape exactly onto itself. Folding a parallelogram along a diagonal or through the middle of a side does not make the halves match.
- A half-turn (180°) about the centre does map the parallelogram onto itself, so it fits twice in a full turn.
- A parallelogram has 0 lines of symmetry and rotational symmetry of order 2
Feb/Mar 2026 · Paper 4 · 0580/42/F/M/26 · Q8
A prism has 6 planes of symmetry. The cross-section of the prism is a regular polygon. Write down the name of the regular polygon.
Show solution
- A prism with a regular n-sided cross-section has n vertical planes of symmetry, plus 1 horizontal plane that cuts the prism in half along its length.
- Total = n + 1 = 6, so n = 5.
- A regular polygon with 5 sides is a pentagon
Oct/Nov 2025 · Paper 4 · 0580/41/O/N/25 · Q1
A quadrilateral has rotational symmetry of order 2, and its two diagonals are its only lines of symmetry. Write down the geometrical name of this quadrilateral.
Show solution
- A kite has only one line of symmetry and a square has four, so neither fits.
- A rhombus has exactly two lines of symmetry, and they are its diagonals.
- A half-turn (180°) about the centre maps a rhombus onto itself, so its rotational symmetry is order 2.
- The quadrilateral is a rhombus
Oct/Nov 2025 · Paper 4 · 0580/42/O/N/25 · Q3
Write down the order of rotational symmetry of a regular decagon.
Show solution
- A regular polygon with n sides fits onto itself n times in one full turn.
- A decagon has 10 sides, so the order of rotational symmetry is 10
May/June 2025 · Paper 2 · 0580/22/M/J/25 · Q1
(a) A 4 by 4 grid has three small squares shaded: row 1 column 2, row 1 column 4 and row 4 column 1. Shade one more small square so that the diagram has one line of symmetry.
(b) A 4 by 4 grid has three small squares shaded: row 1 column 4, row 3 column 1 and row 4 column 1. Shade one more small square so that the diagram has rotational symmetry of order 2.
Show solution
- Look for a mirror line that already works for some of the squares.
- The squares in row 1 column 4 and row 4 column 1 are mirror images in either diagonal of the grid.
- Reflect the remaining square (row 1, column 2) in a diagonal to find the square to shade.
- Shade row 2, column 1 (symmetry in the diagonal from top-left to bottom-right), or row 3, column 4 (symmetry in the other diagonal).
- Order 2 means the pattern looks the same after a half-turn (180°) about the centre of the grid.
- A half-turn sends row r, column c to row 5 − r, column 5 − c.
- Row 1 column 4 ↔ row 4 column 1, which are both shaded already.
- Row 3 column 1 goes to row 2, column 4, which is not shaded yet.
- Shade row 2, column 4.
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