Symmetry

A short topic with two ideas — reflective and rotational symmetry — and a small set of shapes whose properties are worth knowing exactly, because they are what gets tested.

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1. Line (reflective) symmetry

A line of symmetry is a mirror line: the shape on one side is the exact reflection of the other.

The test: if you folded the shape along that line, the two halves would land exactly on top of each other.

Shape	Lines of symmetry
Square	4
Rectangle	2
Rhombus	2 (its diagonals)
Parallelogram	0
Kite	1
Isosceles trapezium	1
Equilateral triangle	3
Isosceles triangle	1
Scalene triangle	0
Circle	infinite

A parallelogram has NO lines of symmetry. This is the most-tested fact in the topic and was got wrong repeatedly — students suggested one or two. Fold a parallelogram along either diagonal and the halves do not match. It has rotational symmetry instead.

A rectangle’s diagonals are NOT lines of symmetry. Its two lines run through the midpoints of opposite sides — horizontal and vertical, not corner to corner. Folding a non-square rectangle along a diagonal does not work.

A rhombus is different: its diagonals are its lines of symmetry, because all four sides are equal.

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2. Rotational symmetry

The order of rotational symmetry is the number of times a shape looks identical during one full 360° turn.

Shape	Order
Square	4
Rectangle	2
Parallelogram	2
Rhombus	2
Kite	1
Equilateral triangle	3
Regular pentagon	5
Regular hexagon	6
Circle	infinite

Every shape returns to its start after 360°, so the minimum order is 1. An order of 1 means no rotational symmetry — it is never 0.

The parallelogram has order 2 even though it has no lines of symmetry. The two types of symmetry are independent.

Angle of rotation = 360° ÷ order. For a square, 360 ÷ 4 = 90°.

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3. Regular polygons

A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

• Equilateral triangle: 3 and 3
• Square: 4 and 4
• Regular pentagon: 5 and 5
• Regular hexagon: 6 and 6
• Regular decagon: 10 and 10

Count the sides first. A decagon has 10 sides, so order 10 — answering 5 was a specific recorded error, as was forgetting how many sides a decagon has. Deca = 10.

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4. Planes of symmetry (3D)

A plane of symmetry slices a solid into two mirror-image halves.

Solid	Planes of symmetry
Cube	9
Cuboid (all edges different)	3
Square-based pyramid	4
Cylinder	infinite (plus one horizontal)
Sphere	infinite
Triangular prism (equilateral)	4

A square-based pyramid has 4 planes of symmetry, not 5 — a documented error. They are: two through the midpoints of opposite base edges, and two through the diagonals of the base. There is no horizontal plane, because the top and bottom halves are not mirror images.

A cuboid has 3 — one parallel to each pair of opposite faces. A cube has those 3 plus 6 diagonal planes, giving 9.

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5. Method

To find lines of symmetry: imagine folding the shape along each candidate line. Try vertical, horizontal, then the diagonals.

To find the order of rotational symmetry: imagine turning the shape through a full circle and count how many times it looks the same, including the starting position.

Tracing paper is allowed in the exam and makes both checks straightforward — trace the shape and physically rotate or flip it.

Completing a symmetrical pattern: reflect each point to the same perpendicular distance on the other side of the mirror line, or rotate each point about the centre.

Shade the correct squares. For rotational symmetry questions on a grid, rotate the pattern about the marked centre rather than reflecting it — mixing the two up was a recorded error.

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6. Mistakes that cost marks

Saying a parallelogram has lines of symmetry — it has none.

Treating a rectangle’s diagonals as lines of symmetry.

Giving order 0 instead of 1.

Confusing the order with the number of lines of symmetry.

Miscounting the sides of a polygon, especially a decagon.

Saying a square-based pyramid has 5 planes.

Reflecting when the question asked for a rotation.

Forgetting the starting position when counting the order.

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Frequently asked questions

What is a line of symmetry? A mirror line — folding along it makes the two halves match exactly.

How many lines of symmetry does a parallelogram have? None.

Are a rectangle’s diagonals lines of symmetry? No. Its lines run through the midpoints of opposite sides.

What is the order of rotational symmetry? The number of times a shape looks identical in a full 360° turn.

Can the order be 0? No — the minimum is 1, meaning no rotational symmetry.

How many lines of symmetry does a regular polygon have? The same as its number of sides — and the same order of rotational symmetry.

What is a plane of symmetry? A flat slice cutting a 3D solid into two mirror-image halves.

How many planes of symmetry does a cube have? 9. A cuboid has 3.

How many does a square-based pyramid have? 4.

Can I use tracing paper? Yes — it is allowed and useful.

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Quick revision checklist

• I can find lines of symmetry by folding
• I know a parallelogram has none
• I know a rectangle’s lines are through midpoints, not diagonals
• I know a rhombus’s diagonals are lines of symmetry
• I can find the order of rotational symmetry, counting the start
• I know the minimum order is 1
• I can calculate the angle of rotation as 360 ÷ order
• I know a regular n-gon has n lines and order n
• I can count sides correctly, including a decagon
• I know the planes of symmetry for a cube, cuboid and square-based pyramid
• I can complete symmetrical patterns by reflection and rotation

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These notes cover symmetry in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons. This was among the least-covered topics in that set, so the page follows the syllabus closely and is deliberately concise rather than padded. Always check the current syllabus for your own exam series.