What is rotational symmetry, and how do you find its order?

Put a pencil point in the middle of a paper windmill and spin it. After a quarter turn, the windmill looks just like it did before. That is rotational symmetry: the shape fits onto itself after a turn of less than a full turn. The point it spins around is the centre of rotation.

The order of rotational symmetry is the number of times a shape looks exactly the same during one full turn (360°). Count the starting position at the end of the turn, but not twice. A square fits onto itself after 90°, 180°, 270° and 360°, so its order is 4.

Every shape fits onto itself after a complete turn, so every shape has order at least 1. That is why order 1 means no rotational symmetry. A shape needs order 2 or more to have rotational symmetry.

Once you know the order, you know the smallest angle of turn:

smallest angle of rotation = 360° ÷ order

A regular pentagon has order 5, so it fits onto itself every 360° ÷ 5 = 72°. It works backwards too: a shape that fits every 120° has order 360 ÷ 120 = 3.

Rotational symmetry is different from line (mirror) symmetry, where you fold a shape so the halves match. A shape can have one without the other. The letter S has rotational symmetry of order 2 but no line of symmetry; a heart has one line of symmetry but no rotational symmetry. Read more about folding and mirror lines on the symmetry page.

A handy pattern: if a flat shape has at least one line of symmetry, its order of rotational symmetry equals its number of lines of symmetry. A square has 4 lines and order 4; an equilateral triangle has 3 lines and order 3; a rectangle has 2 and 2. Every regular polygon with n sides has n lines of symmetry and order n. Turning a shape is one of the transformations you will meet on grids, along with reflection and translation.

No tracing paper? Look for identical parts arranged evenly around the centre, such as the blades of a fan or the arms of a star. If a shape is made of 5 identical arms evenly spaced, its order is 5.

What is rotational symmetry in maths?#

A shape has rotational symmetry if it fits exactly onto itself after a turn of less than 360° about a fixed point, its centre. The number of times it fits in one full turn is its order.

How do you find the order of rotational symmetry?#

Turn the shape (or a tracing of it) once all the way round its centre and count how many times it looks exactly the same. The full turn counts once. For a regular polygon, the order is simply the number of sides.

What is rotational symmetry of order 2?#

It means the shape looks the same twice in a full turn: after a half turn (180°) and after the full turn. Rectangles, parallelograms, rhombuses and the letters S, Z, N and H have order 2. In some countries this is called point symmetry.

Does a triangle have rotational symmetry?#

Only an equilateral triangle does: it has order 3 and fits every 120°. Isosceles and scalene triangles have order 1, which means no rotational symmetry.

What is the rotational symmetry of a pentagon?#

A regular pentagon has rotational symmetry of order 5 and fits onto itself every 72°. An irregular pentagon (sides or angles not all equal) usually has no rotational symmetry.

What is the difference between rotational symmetry and line symmetry?#

Line symmetry is about folding or reflecting: one half is the mirror image of the other. Rotational symmetry is about turning. A square has both; the letter S has only rotational symmetry; a heart has only line symmetry.

Does a circle have rotational symmetry?#

Yes, of infinite order. It looks the same however far you turn it about its centre, and it also has infinitely many lines of symmetry.

What is the line of symmetry of a parabola?#

The graph of y = ax² + bx + c has one vertical line of symmetry through its turning point, x = −b ÷ (2a). For y = x² − 6x + 5 it is x = 3. Parabolas have line symmetry but no rotational symmetry. See quadratic equations.