Shapes and symmetry
Identifying the symmetry of 2D shapes
Two different kinds of symmetry, and a shape can have one without the other. A parallelogram is the proof.
Lines of symmetry
A LINE OF SYMMETRY is a line you could fold along so both halves match exactly. It is also called reflective symmetry. A square has 4. A rectangle has 2. A kite has 1. A parallelogram has none at all.
Rotational symmetry
The ORDER OF ROTATIONAL SYMMETRY is how many times a shape looks exactly the same during one full turn. Every shape looks the same after a full 360° turn, so the order is never less than 1. A rhombus has order 2: it matches once after 180° and again after 360°.
You can have one without the other
A parallelogram has NO lines of symmetry — you cannot fold it to match. But it has order 2 rotational symmetry, because turning it 180° leaves it looking identical. That is the key idea of this section: the two kinds of symmetry are independent.
The quadrilateral table
square — 4 lines, order 4 rectangle — 2 lines, order 2 rhombus — 2 lines, order 2 parallelogram — 0 lines, order 2 kite — 1 line, order 1 trapezium — 0 lines, order 1 isosceles trapezium — 1 line, order 1 And for triangles: scalene — 0 lines, order 1 isosceles — 1 line, order 1 equilateral — 3 lines, order 3 Notice the pattern in the regular shapes: a REGULAR polygon with n sides has n lines of symmetry AND order n. A circle has infinitely many of both.
Worked examples
How many lines of symmetry has a square?
- Two through opposite sides, two through opposite corners.
Answer: 4
A parallelogram's symmetry
- You cannot fold it so both halves match — 0 lines.
- But turning it 180° leaves it looking identical.
Answer: 0 lines, but order 2 rotational symmetry
The order of rotational symmetry of a rhombus
- It matches after 180°, and again after 360°.
Answer: Order 2
Why the order is never less than 1
- Every shape looks the same after a full 360° turn.
Answer: So the minimum order is 1
An equilateral triangle
- Three equal sides give three fold lines.
- It is regular, so the order matches.
Answer: 3 lines, order 3
A regular polygon with n sides
- The pattern holds for every regular shape.
Answer: n lines of symmetry and order n
Where you see this in real life
- A playing card has order 2 rotational symmetry but no line of symmetry — which is exactly why it looks right either way up.
- Road signs use symmetry deliberately: a regular octagon stop sign looks the same from 8 different rotations.
Key words
- Line of symmetry
- a fold line where both halves match exactly
- Order of rotational symmetry
- how many times a shape matches itself in a full turn
- Regular polygon
- a polygon with all sides and all angles equal
- Reflective symmetry
- another name for line symmetry
- Rotational symmetry
- looking the same during a turn
- Quadrilateral
- a shape with four straight sides