Line and Rotational Symmetry (Examples)

A figure has symmetry when a rigid motion maps it exactly onto itself. It has line symmetry if a reflection over a line matches it with itself. It has rotational symmetry if a turn of less than 360° about its center matches it. A regular polygon with $n$ sides has $n$ lines of symmetry.

Before this lesson: You should know Reflections, Rotations and Measuring and Classifying Angles.

What Is Symmetry?

Fold a paper heart down the middle. The two halves land exactly on top of each other. Now spin a pinwheel a little. At some point, it looks just like it did before you spun it. Both are kinds of symmetry.

In geometry, symmetry means a transformation moves the figure, but the figure ends up covering the same spot. It looks like nothing happened.

Line Symmetry

A figure has line symmetry (also called reflection symmetry) when you can reflect it over a line and get the same figure. That line is a line of symmetry. Each point on one side matches a point on the other side, the same distance from the line.

Three shapes in blue, each with its lines of symmetry drawn as dashed green lines. An isosceles trapezoid has 1 line of symmetry, a vertical line through the middles of its two parallel sides. A rectangle has 2 lines of symmetry, one vertical and one horizontal, through the middles of opposite sides. A regular hexagon has 6 lines of symmetry: 3 through opposite vertices and 3 through the middles of opposite sides. — Fold along any dashed line and the two halves match. A rectangle’s diagonals are not lines of symmetry.

A rectangle that is not a square has only 2 lines of symmetry. Its diagonals are not lines of symmetry. If you fold a long rectangle along a diagonal, the corners don’t match.

Rotational Symmetry

A figure has rotational symmetry when a turn of more than 0° and less than 360° about its center maps it onto itself. Every figure matches itself after a full 360° turn, so that one doesn’t count.

The order is the number of times the figure matches itself during one full turn, counting the 360° turn.
The angle of rotational symmetry is the smallest turn that works. It equals $360^\circ \div \text{order}$ .
A figure with 180° rotational symmetry also has point symmetry. Each point has a partner straight through the center.

A regular pentagon matches itself every 72°. A parallelogram matches itself after a half turn.

Symmetry Rules for Common Shapes

For a regular polygon with $n$ sides, use these rules.

\text{lines of symmetry} = n \qquad \text{order} = n \qquad \text{smallest angle} = \frac{360^\circ}{n}

Shape	Lines of symmetry	Rotational symmetry	Smallest angle
Equilateral triangle	3	Order 3	120°
Square	4	Order 4	90°
Rectangle (not a square)	2	Order 2	180°
Rhombus (not a square)	2 (the diagonals)	Order 2	180°
Parallelogram (not a rectangle or rhombus)	0	Order 2	180°
Isosceles trapezoid	1	None	—
Kite (not a rhombus)	1	None	—
Regular pentagon	5	Order 5	72°
Regular hexagon	6	Order 6	60°
Circle	Infinitely many	Any angle	Any angle

How to Find the Symmetry of a Figure

Look for a line that splits the figure into two mirror halves. Check each possible line by folding in your mind.
Count every line that works.
Find the center of the figure.
Turn the figure in your mind. Note every angle less than 360° where it matches itself.
The order is the number of matching turns, plus 1 for the full turn.
On a grid, check with a rule. Reflect or rotate every vertex. If you get the same set of vertices back, the figure has that symmetry.

Worked Examples

Example 1: Compare three quadrilaterals

Find the lines of symmetry and the rotational symmetry of a square, a rectangle that is 6 by 4, and an isosceles trapezoid.

Square: the two lines through the middles of opposite sides work. The two diagonals also work. That makes 4 lines.
Square: it matches itself after 90°, 180°, 270° and 360°. That is order 4.
Rectangle: only the two lines through the middles of opposite sides work. That makes 2 lines.
Rectangle: it matches itself after 180° and 360°. That is order 2.
Isosceles trapezoid: only the line through the middles of the two parallel sides works. That makes 1 line.
Isosceles trapezoid: no turn less than 360° works, so it has no rotational symmetry.

Answer: square: 4 lines, order 4. Rectangle: 2 lines, order 2. Isosceles trapezoid: 1 line, no rotational symmetry.

Example 2: A regular octagon

A stop sign is a regular octagon, with 8 equal sides and 8 equal angles. Find its lines of symmetry, its order and every angle of rotational symmetry.

A regular polygon with $n$ sides has $n$ lines of symmetry. Here $n = 8$ , so it has 8 lines.
The order equals $n$ , so the order is 8.
Find the smallest angle: $360^\circ \div 8 = 45^\circ$
List the multiples of 45° below 360°: 45°, 90°, 135°, 180°, 225°, 270°, 315°

Answer: 8 lines of symmetry, order 8. It matches itself after turns of 45°, 90°, 135°, 180°, 225°, 270° and 315°.

Example 3: Check line symmetry on a grid

A trapezoid has vertices $A(-3, 1)$ , $B(3, 1)$ , $C(2, 4)$ and $D(-2, 4)$ . Is the y-axis a line of symmetry?

Reflect each vertex over the y-axis with $(x, y) \to (-x, y)$ . Then see if you get the same four vertices back.

Vertex	Opposite of x	Image	Is it a vertex?
$A(-3, 1)$	$-(-3) = 3$	$(3, 1)$	Yes, it is B
$B(3, 1)$	$-(3) = -3$	$(-3, 1)$	Yes, it is A
$C(2, 4)$	$-(2) = -2$	$(-2, 4)$	Yes, it is D
$D(-2, 4)$	$-(-2) = 2$	$(2, 4)$	Yes, it is C

Answer: Yes. The reflection swaps A with B and C with D, so the y-axis is a line of symmetry.

Example 4: Check point symmetry on a grid

A parallelogram has vertices $P(1, 2)$ , $Q(4, 2)$ , $R(-1, -2)$ and $S(-4, -2)$ . Does it have 180° rotational symmetry about the origin?

Rotate each vertex 180° with $(x, y) \to (-x, -y)$ .

$P(1, 2) \to (-1, -2)$ , which is R.
$Q(4, 2) \to (-4, -2)$ , which is S.
$R(-1, -2) \to (1, 2)$ , which is P.
$S(-4, -2) \to (4, 2)$ , which is Q.
Every image is a vertex of the same parallelogram.

Answer: Yes. It has rotational symmetry of order 2 (point symmetry) about the origin.

Example 5: Word problem (ceiling fan)

A ceiling fan has 5 identical blades, evenly spaced. What is the smallest turn that makes the fan look the same? Does it look the same after a 144° turn? After a 100° turn?

The blades are evenly spaced, so the order is 5.
Find the smallest angle: $360^\circ \div 5 = 72^\circ$
Check 144°: $144^\circ \div 72^\circ = 2$ . That is a whole number, so yes.
Check 100°: $100^\circ \div 72^\circ$ is not a whole number, so no.

Answer: The smallest turn is 72°. The fan looks the same after 144° but not after 100°.

Common Mistakes

Mistake	Example of the error	Fix
Counting a rectangle’s diagonals	Saying a 6 by 4 rectangle has 4 lines	Fold on a diagonal: the corners miss. It has 2 lines.
Counting the 360° turn as symmetry	Saying a kite has rotational symmetry	Only turns less than 360° count.
Mixing up order and angle	Saying a hexagon has order 60	Order is a count (6). The angle is $360^\circ \div 6 = 60^\circ$ .
Giving a parallelogram a line of symmetry	Drawing a line through the middle of a slanted parallelogram	A general parallelogram has 0 lines, but order 2.
Missing half the lines of a regular polygon	Saying a regular hexagon has 3 lines	Count lines through vertices and through the middles of sides: 6.

Symmetry in Letters and Graphs

Symmetry shows up in letters, logos and graphs.

The letter H has 2 lines of symmetry and order 2. The letter N has no lines of symmetry but order 2. The letter A has 1 line and no rotational symmetry.
The graph of $y = x^2$ is symmetric about the y-axis. The graph of $y = x^3$ has point symmetry about the origin. You can see both in the parent functions lesson.
A figure with both a horizontal and a vertical line of symmetry always has 180° rotational symmetry. Two reflections over crossing lines make a rotation, as shown in compositions of transformations.

Practice Problems

How many lines of symmetry does an equilateral triangle have?
Find the order and the smallest angle of rotational symmetry of a regular hexagon.
Find the smallest angle of rotational symmetry of a regular decagon (10 sides).
Does a kite (not a rhombus) have rotational symmetry? How many lines of symmetry does it have?
A regular polygon has a smallest angle of rotational symmetry of 40°. How many sides does it have?
Describe the symmetry of the letters H, N and A.
A rhombus has vertices $(-4, 0)$ , $(0, 3)$ , $(4, 0)$ and $(0, -3)$ . Find its lines of symmetry. Does it have rotational symmetry?
A rectangle has vertices $(1, 1)$ , $(5, 1)$ , $(5, 3)$ and $(1, 3)$ . Is $x = 3$ a line of symmetry? Is $y = 2$ ?
A regular polygon has rotational symmetry of order 12. Find its smallest angle and its number of lines of symmetry.
A logo is a star with 8 identical points. What is the smallest turn that makes it look the same? How many times does it match itself in one full turn?

Show answers

An equilateral triangle is a regular polygon with $n = 3$
It has 3 lines of symmetry, one from each vertex to the middle of the opposite side.
Order: $n = 6$
Smallest angle: $360^\circ \div 6 = 60^\circ$
$360^\circ \div 10 = 36^\circ$
No rotational symmetry. Only the full 360° turn matches.
It has 1 line of symmetry, the line through the two vertices where the equal sides meet.
Use $\text{angle} = 360^\circ \div n$
So $n = 360^\circ \div 40^\circ$
Divide: $n = 9$
It has 9 sides.
H: 2 lines (vertical and horizontal), order 2
N: 0 lines, order 2
A: 1 line (vertical), no rotational symmetry
Reflect over the y-axis with $(-x, y)$ : $(-4, 0) \leftrightarrow (4, 0)$ , and $(0, 3)$ and $(0, -3)$ stay put. Yes.
Reflect over the x-axis with $(x, -y)$ : $(0, 3) \leftrightarrow (0, -3)$ , and $(-4, 0)$ and $(4, 0)$ stay put. Yes.
Lines of symmetry: the x-axis and the y-axis
Rotate 180° with $(-x, -y)$ : each vertex lands on the opposite vertex, so it has order 2.
Reflect over $x = 3$ with $(6 - x, y)$ : $(1, 1) \to (5, 1)$ and $(1, 3) \to (5, 3)$ , so the vertices swap. Yes.
Reflect over $y = 2$ with $(x, 4 - y)$ : $(1, 1) \to (1, 3)$ and $(5, 1) \to (5, 3)$ , so the vertices swap. Yes.
Both are lines of symmetry.
Smallest angle: $360^\circ \div 12 = 30^\circ$
A regular polygon has as many lines as its order: 12 lines
Smallest turn: $360^\circ \div 8 = 45^\circ$
It matches itself 8 times in one full turn (order 8).

FAQ

What is the difference between line symmetry and rotational symmetry?

Line symmetry means a reflection over a line maps the figure onto itself. Rotational symmetry means a turn of less than 360° does.

How do you find the order of rotational symmetry?

Count how many times the figure matches itself in one full turn, including the 360° turn. For a regular polygon, the order equals the number of sides.

How many lines of symmetry does a rectangle have?

A rectangle that is not a square has 2 lines of symmetry. A square has 4.

What is point symmetry?

Point symmetry is rotational symmetry of 180°. Every point has a matching point the same distance away on the opposite side of the center.

Summary

A figure has symmetry when a rigid motion maps it onto itself.
Line symmetry: a reflection over a line of symmetry matches the figure.
Rotational symmetry: a turn of less than 360° matches the figure.
Smallest angle of rotational symmetry = $360^\circ \div \text{order}$ .
A regular polygon with $n$ sides has $n$ lines of symmetry and order $n$ .
On a grid, check symmetry by reflecting or rotating every vertex.

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