Symmetry

Line (reflective) symmetry

Look closely at a snowflake, a flower's petals, or the grand front of a cathedral and you spot the same thing: the left half is a perfect mirror of the right. That balance is symmetry, and we notice it everywhere — it's why a face drawn lopsided looks "wrong", and why designers and builders reach for it to make things feel steady and pleasing.

A shape has a line of symmetry if you could fold it along that line and the two halves would land exactly on top of each other — no overhang, no gap. The fold line acts like a mirror: everything on one side is reflected onto a perfect match on the other. This is also called reflective symmetry.

Try it in your head with a butterfly: fold it down the middle and the left wing covers the right wing perfectly. That middle crease is its line of symmetry. A shape that has at least one such line is called symmetrical.

Cut any shape out of paper and try to fold it so one half sits exactly on the other. If you can, your fold is a line of symmetry — trace along the crease to draw it. If no fold ever lines the halves up, the shape has no line of symmetry. A scrunched, lopsided blob is the classic example: there is simply no fold that works.

How many lines? Some have several, some none

A shape can have more than one line of symmetry — or none at all. Counting them is just asking: how many different folds line the halves up?

A square has 4 lines of symmetry: one through each pair of opposite edges (up–down and left–right), and one along each diagonal.
A rectangle (that is not a square) has only 2 — the up–down fold and the left–right fold. (Its diagonals do not work; see below.)
An equilateral triangle has 3, an isosceles triangle has just 1, and a scalene triangle (all sides different) has 0.
A circle has infinitely many — every line through its centre folds it perfectly.

In general a regular n-gon has exactly n lines of symmetry: a regular pentagon has 5, a regular hexagon has 6, and so on.

A regular five-pointed star has 5 lines of symmetry — one straight down through each point to the dip between the two opposite points. Fold along any of those five lines and the star matches itself exactly. (A six-pointed star would have 6.)

Worked examples

A square. Fold the top down onto the bottom — they match: that is one line. Fold the left onto the right — match: a second. Now fold corner-to-corner along a diagonal — the two triangles match: a third. The other diagonal gives a fourth. No other fold works, so a square has exactly 4 lines of symmetry.

The letter A. Fold it left-to-right down the middle and the two slanted legs swap places perfectly: 1 line of symmetry. Fold it top-to-bottom and the pointed top does not match the open legs, so there is no horizontal line.

A rectangle. Up–down works, left–right works — that is 2. Tempting as it looks, a diagonal fold does not match the halves (a long thin rectangle folded corner-to-corner spills over the edge). So a non-square rectangle has exactly 2 lines, not 4.

A diagonal is not always a line of symmetry. It is for a square, but for a rectangle the diagonal fold fails — the halves do not land on each other. Always check by folding, never assume.
"Cutting into two equal halves" is not enough. A line can split a shape into two pieces of equal area yet the pieces still not be mirror images. A line of symmetry needs the halves to be reflections of each other.

See the lines

Step through the figure: first a square with its four mirror lines, then an equilateral triangle with its three, and finally the rotational order of each.

Rotational symmetry

There is a second kind of symmetry. A shape has rotational symmetry if you can turn it about its centre by less than a full turn and it looks exactly the same as before. The order m counts how many times the shape matches itself during one complete 360^\circ turn.

A square has rotational symmetry of order 4 — it looks the same after every 90^\circ — and an equilateral triangle has order 3, matching after every 120^\circ. A regular n-gon has rotational symmetry of order n. A shape can have rotational symmetry with no line of symmetry at all — a pinwheel or a recycling logo spins onto itself but cannot be folded to match.

line (reflective) symmetry — a fold line that maps the shape exactly onto itself;
rotational symmetry — how many times a turn maps the shape onto itself in one full 360^\circ (its order);
a regular n-gon has n lines of symmetry and rotational symmetry of order n.

Symmetry in letters and in nature

Symmetry is everywhere once you look. Many capital letters have a line of symmetry: A, M, T, U, V, W, Y have a vertical one; B, C, D, E, K have a horizontal one; and H, I, O, X have both. The letters F, G, J, L, P, Q, R have none.

Nature loves the mirror too. A butterfly, a leaf, a snowflake (which has 6 lines), and an animal's face all show line symmetry — the left side mirrors the right.

owl face Look at this owl head-on. Draw a line straight down the middle — between the eyes, down the beak — and the left half mirrors the right: one eye on each side, one ear-tuft on each side. That single vertical fold is its line of symmetry. Almost every animal face works this way, which is why a face drawn lopsided looks "wrong".

Spot the lines yourself

Here is a fresh example each time. A random shape or letter appears with every line of symmetry drawn as a dashed mirror line — fold along any dashed line and the two halves match. Some shapes show several lines; a few show none. Press Refresh for a new one.