Symmetry: Line & Rotational

A figure has line symmetry if you can fold it along a line so the two halves land exactly on each other. That fold line is called a line of symmetry, and it splits the figure into two mirror-image halves. A figure has rotational symmetry if you can turn it less than $360^\circ$ about its center and it looks exactly the same as before.

Both kinds of symmetry come down to counting: how many fold lines a figure has, and how small a turn brings it back onto itself. Regular polygons make both counts predictable, and that pattern handles most test questions.

Line symmetry and the fold test

To test a line, imagine folding the figure along it. If every point lands on a matching point — corners on corners, edges on edges — it is a line of symmetry. If anything hangs over, it is not.

The triangle below is isosceles: its two slanted sides are the same length. The dashed line through the top vertex and the midpoint of the base is its only line of symmetry — folding there matches the two equal sides. No other fold works, so this triangle has exactly $1$ line of symmetry.

A regular polygon — all sides equal, all angles equal — has exactly as many lines of symmetry as it has sides. An equilateral triangle has $3$ , a square has $4$ , a regular hexagon has $6$ .

Rotational symmetry

Spin a figure about its center. If it matches its original position after a turn of less than $360^\circ$ , it has rotational symmetry. The angle of rotational symmetry is the smallest such turn.

When a figure matches itself $n$ times during one full turn, that smallest angle is $\dfrac{360^\circ}{n}$ . A regular pentagon matches itself $5$ times, so its angle of rotational symmetry is $360 \div 5 = 72^\circ$ . The number of matches, $n$ , is called the order of rotational symmetry.

Every figure matches itself after a full $360^\circ$ turn, so that turn never counts. If the only turn that works is $360^\circ$ , the figure has no rotational symmetry.

Counting quickly on common figures

Regular polygon with $n$ sides: $n$ lines of symmetry, and rotational symmetry of $\dfrac{360^\circ}{n}$ . That single fact answers most symmetry questions.

Non-square rectangle: $2$ lines of symmetry — one through the midpoints of the long sides, one through the midpoints of the short sides. The diagonals are not lines of symmetry, because folding along a diagonal leaves the unequal sides mismatched. It does have $180^\circ$ rotational symmetry.

Scalene triangle: no equal sides, so $0$ lines of symmetry and no rotational symmetry. Isosceles (non-equilateral) triangle: exactly $1$ line of symmetry, no rotational symmetry.

Worked examples

Example 1: lines of symmetry of a regular pentagon

How many lines of symmetry does a regular pentagon have?

A regular polygon has as many lines of symmetry as sides

n = 5

Each line runs from a vertex to the midpoint of the opposite side

Count them

5

Answer: $5$ lines of symmetry

Example 2: angle of rotational symmetry

Find the angle of rotational symmetry of a regular octagon.

A regular octagon matches itself

8

times in a full turn

n = 8

Divide a full turn by the number of matches

\dfrac{360^\circ}{8} = 45^\circ

Answer: $45^\circ$

Example 3: a non-square rectangle

How many lines of symmetry does a rectangle that is not a square have, and does it have rotational symmetry?

The fold through the midpoints of the long sides works

1

The fold through the midpoints of the short sides works

1 + 1 = 2

A diagonal fold leaves the corners mismatched, since the side lengths differ

A half turn maps the rectangle onto itself

180^\circ

Answer: $2$ lines of symmetry, and rotational symmetry of $180^\circ$

Try one yourself

Common questions

Can a figure have rotational symmetry but no line symmetry?

Yes. A pinwheel shape and a parallelogram (that is not a rectangle or rhombus) both map onto themselves under a rotation but have no fold line that works. The two kinds of symmetry are independent — check each one separately.

Does the full 360-degree turn count as rotational symmetry?

No. Every figure looks the same after a complete turn, so that would make the idea meaningless. Rotational symmetry requires a match at some angle strictly less than $360^\circ$ .

What is the order of rotational symmetry?

The number of times a figure matches itself during one full turn. A square has order $4$ because it matches at $90^\circ$ , $180^\circ$ , $270^\circ$ , and $360^\circ$ . Order and angle are linked: angle $= 360^\circ \div$ order.

Why aren't the diagonals of a rectangle lines of symmetry?

Fold a non-square rectangle along a diagonal and the two triangular halves do not line up — a long side lands on a short side. Only the square, where all four sides are equal, has diagonals that work as lines of symmetry.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Symmetry: Line & Rotational

Line symmetry and the fold test

Rotational symmetry

Counting quickly on common figures

Worked examples

Example 1: lines of symmetry of a regular pentagon

Example 2: angle of rotational symmetry

Example 3: a non-square rectangle

Try one yourself

Common questions

Keep learning

Reflections

Translations

Compositions of Transformations

Rotations