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Year 5 Maths 2D Shapes, Position & Transformations

Line & rotational symmetry

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Theory

A shape has line symmetry if a line can fold it so the two halves match exactly. It has rotational symmetry if it fits onto itself more than once as it turns a full turn about its centre; the number of times it fits is its order of rotational symmetry. This Year 5 maths page shows how to find both for squares, rectangles, triangles, regular polygons, other quadrilaterals and letters.

A line of symmetry (mirror line) divides a shape into two halves that are mirror images. If you folded along it, the halves would match exactly.

The order of rotational symmetry is the number of times a shape looks exactly the same during one full turn about its centre. Every shape fits at least once (at the end of the full turn), so the smallest order is \(1\), which means no rotational symmetry.

A regular polygon has all sides equal and all angles equal. A regular polygon with \(n\) sides has \(n\) lines of symmetry and rotational symmetry of order \(n\).

Not every line through the middle is a line of symmetry. A diagonal of a rectangle (that is not a square) cuts it into two triangles that do not match when folded.

Lines of symmetry of a square, rectangle and equilateral triangle Three shapes with their lines of symmetry dashed in red: a square with 4 lines (two through the middles of the sides and two diagonals), a rectangle with 2 lines (through the middles of opposite sides) and an equilateral triangle with 3 lines (each from a corner to the middle of the opposite side). square: 4 rectangle: 2 equilateral: 3
Lines of symmetry: square 4, rectangle 2, equilateral triangle 3.
Parallelogram, isosceles trapezium and regular hexagon A parallelogram with its centre marked and no lines of symmetry; an isosceles trapezium with one vertical line of symmetry; and a regular hexagon with its centre marked, which fits onto itself 6 times in a full turn. parallelogram: 0 trapezium: 1 hexagon: order 6
A parallelogram has no lines of symmetry but rotational symmetry of order 2. A regular hexagon has order 6.

Symmetry of common shapes:

ShapeLines   Order
square\(4\)    \(4\)
rectangle\(2\)    \(2\)
rhombus\(2\)    \(2\)
parallelogram\(0\)    \(2\)
kite\(1\)    \(1\)
equilateral triangle\(3\)    \(3\)
isosceles triangle\(1\)    \(1\)
regular \(n\)-gon\(n\)    \(n\)
circleinfinitely many, any amount of turn
Trace and turn. Trace the shape, put a pencil point on the centre and turn the tracing. Count how many times it fits before you get back to the start.
Fold test. Imagine folding along the line. If the two halves would not lie exactly on top of each other, it is not a line of symmetry.

How to find lines of symmetry

  1. Try lines through corners and through the middles of sides.
  2. Fold (or imagine folding): do the halves match exactly?
  3. Count every line that works.

How to find the order of rotational symmetry

  1. Mark the centre and one corner.
  2. Turn the shape a full turn about the centre.
  3. Count the positions where it looks exactly the same as at the start (including the end).
Example 1 — Isosceles trapezium
How many lines of symmetry does the isosceles trapezium have, and what is its order of rotational symmetry?
Solution

1 line of symmetry (up the middle). It only fits onto itself after a full turn, so its order is 1.

Example 2 — Letter N
Explain why the capital letter N has no lines of symmetry but has rotational symmetry of order 2.
Solution

Folding along any line leaves the slanted stroke not matching. But a half turn about the centre puts the N exactly back on itself, so it fits twice in a full turn.

Example 3 — Regular decagon
A regular decagon has 10 equal sides and 10 equal angles. How many lines of symmetry does it have, and what is its order of rotational symmetry?
A regular decagon A regular decagon with its centre marked. regular decagon
Solution

10 lines and order 10, as for every regular polygon: the number matches the number of sides.

Example 4 — Letter H
The capital letter H. How many lines of symmetry does it have, and what is its order of rotational symmetry?
Solution

2 lines (one across, one down the middle). It looks the same after a half turn, so its order is 2.

Example 5 — Letter Z
The capital letter Z. How many lines of symmetry does it have, and what is its order of rotational symmetry?
Solution

0 lines, but it fits onto itself after a half turn, so its order is 2.

Example 6 — Letter A
The capital letter A. How many lines of symmetry, and what order of rotational symmetry?
Solution

1 line (down the middle) and order 1 (no rotational symmetry).

Example 7 — Find the mistake
Ivy says a rectangle has 4 lines of symmetry, like a square, because its diagonals cut it in half. Explain the mistake.
Solution

The diagonals cut a rectangle into two equal triangles, but folding along a diagonal does not make them match. A rectangle has only 2 lines of symmetry.

Common pitfalls

Counting the diagonals of a rectangle. They halve the area but are not mirror lines.
Saying order 0. The smallest order of rotational symmetry is 1 (fits only after a full turn).
Mixing up lines and order. The letter N has 0 lines of symmetry but order 2.
Forgetting the final position. When counting the order, the position after the full turn counts as one.

Frequently asked questions

What is a line of symmetry?

A line that folds a shape into two halves that match exactly.

What is the order of rotational symmetry?

The number of times a shape fits onto itself in one full turn about its centre.

How many lines of symmetry does a regular polygon have?

The same as its number of sides. A regular decagon has \(10\) lines and order \(10\).

Can a shape have rotational symmetry but no line symmetry?

Yes. The letter N has no lines of symmetry but rotational symmetry of order 2.

How many lines of symmetry does a circle have?

Infinitely many: every line through the centre is a line of symmetry.

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