Rotational Symmetry

A shape has rotational symmetry when it looks the same after being rotated less than a full turn about a fixed centre point. Explore the order of rotational symmetry and point (central) symmetry.

What Is Rotational Symmetry?

A shape has rotational symmetry when you can rotate it around a fixed central point by less than 360° and it looks exactly the same as it did before the rotation.

The fixed point is called the centre of rotation.

Tip

Trace the shape onto tracing paper, place a pin at the centre, and spin the tracing paper. Every position where the traced shape lines up with the original is a position of symmetry.

Order of Rotational Symmetry

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

Angle of Rotation

\text{angle} = \dfrac{360°}{\text{order}}

Divide 360° by the order of symmetry to find the smallest angle that maps the shape onto itself.

Shape	Order	Smallest angle
Circle	Infinite	Any angle
Square	4	90°
Equilateral triangle	3	120°
Rectangle	2	180°
Regular pentagon	5	72°
Regular hexagon	6	60°
Scalene triangle	1	360° (no rotational symmetry)

Every shape has order at least 1, because a full 360° rotation always brings it back to the start.

Point Symmetry (Central Symmetry)

Point symmetry (also called central symmetry) is a special case of rotational symmetry with order 2 — the shape looks the same after a rotation of exactly 180°.

For every point $A$ on the shape, there is a corresponding point $A'$ on the opposite side of the centre $O$ , at the same distance:

Central Symmetry Rule

OA = OA' \quad \text{and} \quad A, O, A' \text{ are on one straight line}

The centre O is the midpoint of segment AA′. Every point and its image are equally far from O on opposite sides.

Shapes that have point symmetry include: rectangles, squares, parallelograms, circles, all regular polygons with an even number of sides, and the letters S, Z, and N.

Constructing the Image Under Central Symmetry

1
Draw a line from the point through the centre
Start at point A, draw a straight line through centre O, and continue beyond O.
2
Measure the distance OA
Use a ruler or count grid squares to find how far A is from O.
3
Mark A′ on the opposite side
Place A′ at the same distance from O on the other side of the line.
4
Repeat for all key points
Do the same for every vertex of the shape, then connect the image points in order.

Does a regular hexagon have rotational symmetry?

Yes. A regular hexagon has rotational symmetry of order 6.

It maps onto itself after rotations of: 60°, 120°, 180°, 240°, 300°, and 360°.

Because the order is 6 (which includes 180°), the hexagon also has point symmetry.

Finding the image of a point under central symmetry

Centre of symmetry is $O = (0, 0)$ and point $A = (3, -2)$ .

To find $A'$ : change the sign of both coordinates.

$A' = (-3, 2)$

Check: $O$ is the midpoint of $AA'$ — the midpoint of $(3, -2)$ and $(-3, 2)$ is $\left(\frac{3+(-3)}{2}, \frac{-2+2}{2}\right) = (0, 0)$ . ✓

Real world: playing cards

A standard playing card (for example, the 6 of hearts) looks the same when rotated 180° — the symbols appear both right-side-up and upside-down. This is point symmetry used deliberately so that either player at the table can read the card.

Real world: wheels and snowflakes

A wheel with 6 spokes has rotational symmetry of order 6 — it looks identical every 60°. This is why wheels work smoothly: every part is balanced.

A snowflake also has rotational symmetry of order 6. All natural snowflakes are hexagonal due to the structure of ice crystals.

Common Mistakes

Counting the full 360° rotation as an extra position — a full turn always maps any shape onto itself and is not counted separately.
Confusing line symmetry with rotational symmetry — a parallelogram has rotational symmetry (order 2) but no line symmetry.
Forgetting that order 1 means no rotational symmetry — if the only matching position is the full 360° turn, the shape has no rotational symmetry.
Placing the image point on the wrong side of the centre — A and A′ must be on exactly opposite sides of O.

Practice Problems

A regular pentagon is rotated about its centre. After how many degrees does it first look the same as at the start?

Show answer

360° ÷ 5 = 72°. The pentagon first looks identical after a 72° rotation.

Point A is at (4, −2) and the centre of symmetry is at the origin O = (0, 0). What are the coordinates of A′?

Show answer

A′ = (−4, 2). Under central symmetry through the origin, both coordinates change sign.

Does a rectangle have point symmetry? Explain.

Show answer

Yes. A rectangle has rotational symmetry of order 2 (it maps onto itself after 180°), so it has point symmetry. Its centre is the intersection of its diagonals.