Math Core

A butterfly, a snowflake and a stop sign all look balanced, and transformations explain exactly why. A figure has symmetry when some rigid motion other than "do nothing" maps the figure onto itself. The kind of motion tells you the kind of symmetry.

Line symmetry

Definition

Line symmetry

A figure has line symmetry (or reflectional symmetry) if a reflection over some line maps the figure onto itself. That line is a line of symmetry.

If you could fold the figure along a line of symmetry, the two halves would match exactly. On the coordinate plane, you can test a line of symmetry with the reflection rules: every vertex must land on a vertex of the same figure.

Worked example: Testing a line of symmetry

Kite $ABCD$ has vertices $A(1, 4)$ , $B(3, 1)$ , $C(1, -3)$ and $D(-1, 1)$ . Show that the line $x = 1$ is a line of symmetry.

Reflect each vertex over $x = 1$ using $(x, y) \rightarrow (2 - x,\ y)$ :

$A(1, 4) \rightarrow (1, 4) = A$ , since $A$ is on the line.
$B(3, 1) \rightarrow (-1, 1) = D$ .
$C(1, -3) \rightarrow (1, -3) = C$ .
$D(-1, 1) \rightarrow (3, 1) = B$ .

The reflection sends the set of vertices to itself (it swaps $B$ and $D$ ), so it maps the kite onto itself. The line $x = 1$ is a line of symmetry.

Kite ABCD and its line of symmetry, x = 1.Open in grapher →

Rotational symmetry

Definition

Rotational symmetry

A figure has rotational symmetry if a rotation about its center by an angle strictly between $0^\circ$ and $360^\circ$ maps the figure onto itself. The number of positions in one full turn where the figure looks the same is its order of rotational symmetry.

A figure that looks the same in $n$ positions during a full turn has order $n$ , and its smallest angle of rotational symmetry is $\dfrac{360^\circ}{n}$ . Every multiple of that angle also works.

A figure with $180^\circ$ rotational symmetry has point symmetry. For every point $Q$ of the figure, the point directly opposite through the center is also on the figure, and the center is the midpoint between them.

Regular polygons

A regular polygon with $n$ sides has

$n$ lines of symmetry, and
rotational symmetry of order $n$ , with smallest angle $\dfrac{360^\circ}{n}$ .

Worked example: A regular hexagon

Describe the symmetries of a regular hexagon.

It has $6$ lines of symmetry: $3$ through pairs of opposite vertices and $3$ through midpoints of opposite sides. Its smallest angle of rotational symmetry is $\dfrac{360^\circ}{6} = 60^\circ$ , so the rotations that map it onto itself are $60^\circ$ , $120^\circ$ , $180^\circ$ , $240^\circ$ and $300^\circ$ . Since $180^\circ$ is on the list, it also has point symmetry.

Symmetry in quadrilaterals

Symmetry is a quick way to tell the special quadrilaterals apart.

quadrilateral	lines of symmetry	rotational symmetry
parallelogram (general)	$0$	order $2$ ( $180^\circ$ )
rectangle	$2$ (through midpoints of opposite sides)	order $2$
rhombus	$2$ (the diagonals)	order $2$
square	$4$	order $4$ ( $90^\circ$ )
isosceles trapezoid	$1$	none
kite	$1$ (the diagonal joining the vertices where equal sides meet)	none

Common mistake

A diagonal is not always a line of symmetry. Folding a rectangle along a diagonal does not line up the halves unless the rectangle is a square. Test by reflecting: each vertex must land on another vertex.

Symmetry on the coordinate plane

Symmetry lets you complete a figure from part of it. If a figure is symmetric over a line, reflect known points over that line. If it has point symmetry about $M$ , use the midpoint shortcut: the point opposite $Q$ is $2M - Q$ .

Worked example: Completing a figure

A figure has $180^\circ$ rotational symmetry about $M(2, -1)$ , and $Q(5, 3)$ is on it. Find another point that must be on the figure.

The matching point is $2M - Q = (4 - 5,\ -2 - 3) = (-1, -5)$ . Check: the midpoint of $(5, 3)$ and $(-1, -5)$ is $(2, -1)$ . ✓

Graphs of functions can be symmetric too. The parabola $y = (x - 2)^2 - 3$ is symmetric over its axis, the vertical line $x = 2$ : the points $(0, 1)$ and $(4, 1)$ are both on it, equally far from $x = 2$ .

The parabola y = (x − 2)² − 3 is symmetric over the dashed line x = 2.Open in grapher →

Tip

To count rotational symmetry, mark one corner of the figure and imagine turning it. Count how many times the figure fits back into its outline before the marked corner returns home. That count is the order.

Practice

Practice 1

How many lines of symmetry does a regular octagon have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the smallest positive angle, in degrees, of a rotation that maps a regular pentagon onto itself?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

How many rotations with angles strictly between $0^\circ$ and $360^\circ$ map a regular hexagon onto itself?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Which figure has point symmetry but no line of symmetry?

Practice 5

A regular polygon maps onto itself under a rotation of $24^\circ$ , and no smaller positive rotation works. How many sides does it have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A figure is symmetric over the line $x = 3$ . The point $(1, 5)$ is on the figure. Which other point must be on the figure?

Enter a point like (2, -3)

Practice 7

A figure has point symmetry about $M(-1, 2)$ , and $(2, 6)$ is on the figure. Which other point must be on the figure?

Enter a point like (2, -3)

Practice 8

Triangle $ABC$ has vertices $A(0, 5)$ , $B(4, -3)$ and $C(-4, -3)$ . Which statement about its symmetry is true?