Symmetry

What is Symmetry?

Symmetry describes when a shape or pattern looks the same after a transformation — either a reflection or a rotation. Understanding symmetry helps in geometry, art, science, and everyday life.

There are two main types of symmetry in 2D shapes:

1. Line (reflection) symmetry
2. Rotational symmetry

Line (Reflection) Symmetry

A shape has line symmetry if it can be folded along a line so that one half fits exactly on top of the other. The fold line is called the line of symmetry (or mirror line or axis of symmetry).

Key points:

• Each point on one side of the line has a mirror image on the other side, the same distance from the line
• A shape can have more than one line of symmetry
• A regular polygon with n sides has n lines of symmetry

Shape	Lines of Symmetry
Equilateral triangle	3
Square	4
Rectangle	2
Isosceles triangle	1
Parallelogram	0
Regular pentagon	5
Regular hexagon	6
Circle	Infinite
Scalene triangle	0

Drawing Lines of Symmetry

To find a line of symmetry:

1. Imagine folding the shape along a possible line
2. Check whether the two halves match exactly
3. If yes, it is a line of symmetry — mark it with a dashed line

For letters: A, M, T, U, V, W, Y have 1 vertical line; H, I, O, X have 2 lines.

Rotational Symmetry

A shape has rotational symmetry if it looks identical after being rotated by less than 360° about its centre.

The order of rotational symmetry is the number of times the shape looks the same during one full 360° rotation.

Every shape has at least order 1 — it matches itself after a full 360° turn. We say a shape has "no rotational symmetry" when the order is 1 (it only matches at the start/end position).

Shape	Order of Rotational Symmetry	Angle of rotation
Square	4	90°
Equilateral triangle	3	120°
Rectangle	2	180°
Regular pentagon	5	72°
Regular hexagon	6	60°
Parallelogram	2	180°
Scalene triangle	1	360° only
Circle	Infinite	Any angle

Angle formula: angle of rotation = 360° ÷ order

Regular Polygons: A Pattern

For a regular polygon with n sides:

• Lines of symmetry = n
• Order of rotational symmetry = n
• Angle of rotational symmetry = 360° ÷ n

So a regular hexagon (n = 6): 6 lines of symmetry, order 6, rotates by 60°.

Symmetry in Coordinates

Reflection in the y-axis: (x, y) → (-x, y)

• Point (4, -3) reflects to (-4, -3)

Reflection in the x-axis: (x, y) → (x, -y)

• Point (4, -3) reflects to (4, 3)

Reflection in y = x: (x, y) → (y, x)

• Point (4, -3) reflects to (-3, 4)

If a shape is symmetric about the y-axis, for every vertex (x, y) there is a corresponding vertex at (-x, y).

Shapes with Only Rotational Symmetry

A parallelogram (non-rectangular) has:

• Order 2 rotational symmetry (looks the same after 180°)
• 0 lines of symmetry

This shows that rotational symmetry and line symmetry are independent — a shape can have one without the other.

Using Symmetry to Solve Problems

Symmetry reduces the work needed in problems:

• In symmetric shapes, corresponding sides and angles are equal — so you only need to calculate once
• Symmetric diagrams often let you find unknown angles without calculating every angle individually
• In graphs, symmetry about a vertical line means the function is even: f(-x) = f(x)

Common Mistakes

• Confusing the number of lines of symmetry with the order of rotational symmetry (for regular polygons they are equal, but this is not always the case)
• Saying a shape has "no symmetry" when it has order 1 rotational symmetry — order 1 means no rotational symmetry
• Forgetting that a rectangle has only 2 lines of symmetry (not 4 — diagonals are not lines of symmetry for rectangles)
• Counting a line of symmetry twice by drawing both a horizontal and vertical line when only one exists

Tips and Tricks

• For any regular n-gon: lines of symmetry = order of rotation = n
• Use tracing paper to check rotational symmetry — trace the shape, pin the centre, and rotate
• A parallelogram looks like a rectangle "pushed over" — it has rotational symmetry of order 2 but no reflection symmetry
• The letter S has rotational symmetry of order 2 but no line symmetry; the letter Z is the same