Geometry · Lesson 11 of 18

Line and Rotational Symmetry

A shape has line symmetry if a mirror line splits it into two matching halves. It has rotational symmetry if it fits onto itself before a full turn.

Primary
Updated 14 November 2026

What you will learn

Find the lines of symmetry of a shape.
Find the order of rotational symmetry.
Compare the symmetry of common shapes.

Before you start: you should know the names of triangles and quadrilaterals. See Triangles and Quadrilaterals and Their Properties.

The idea

A shape has line symmetry when one straight line, a mirror line, splits it into two halves that are mirror images. Fold along the line and the halves land exactly on each other.

Dashed lines are the mirror lines. A square has 4, a rectangle has 2 and a regular pentagon has 5.

A shape has rotational symmetry when you can turn it about its centre by less than a full turn and it looks exactly the same. The order is the number of times it fits onto itself during one full turn. A shape that only fits after a full turn has order $1$ , which means no rotational symmetry.

A square fits onto itself after every quarter turn, so its order is 4. A rectangle fits after every half turn, so its order is 2.

A table of common shapes

An isosceles triangle and a kite have 1 line each. A rhombus has 2 lines and order 2. A parallelogram has no lines but has order 2.

For any regular polygon with $n$ sides, the number of mirror lines and the order of rotational symmetry are both $n$ .

Worked examples

Example 1: Equilateral triangle

Find the lines of symmetry and the order of rotational symmetry of an equilateral triangle.

Each corner has a line to the middle of the opposite side: $3$ lines.
It fits onto itself every $360 \div 3 = 120^\circ$ : order $3$ .

Answer: $3$ lines, order $3$

Example 2: Rectangle

Find the symmetry of a rectangle that is not a square.

Lines: one joining the midpoints of the long sides and one joining the midpoints of the short sides. The diagonals are not lines of symmetry.
Rotation: a half turn fits it onto itself, so order $2$ .

Answer: $2$ lines, order $2$

Example 3: Letters

Which of the letters H, N and T has both line and rotational symmetry?

H has $2$ lines and fits onto itself after a half turn: both.
N has no line symmetry but fits after a half turn.
T has $1$ line and no rotational symmetry.

Answer: H

Tip: to check a line of symmetry, imagine folding. To check rotation, imagine pinning the centre and turning the shape.

Why it matters

Symmetry shows up in logos, flags, buildings, snowflakes and flowers. Designers use it for balance, and mathematicians use it to classify shapes and to simplify problems, since one half of a symmetric shape tells you about the other.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.