Symmetry: Line and Rotational

By the end of this lesson, you will be able to identify all lines of symmetry in a figure, determine the order of rotational symmetry, and apply symmetry to solve geometric problems.

The Big Idea

Symmetry is a transformation that maps a figure onto itself. A figure has line symmetry if it can be folded along a line so both halves match exactly. It has rotational symmetry if it maps onto itself under a rotation of less than 360°. Symmetry is fundamental to art, architecture, biology, and physics — and it simplifies many geometric arguments dramatically.

Line Symmetry

A line of symmetry (axis of symmetry) divides a figure into two mirror-image halves.

• If you reflect the figure over the line, it maps onto itself.
• A figure can have 0, 1, 2, or infinitely many lines of symmetry.

Lines of Symmetry for Common Figures

Figure	Lines of symmetry
Scalene triangle	0
Isosceles triangle	1 (the altitude from the vertex angle)
Equilateral triangle	3
Rectangle (non-square)	2
Rhombus (non-square)	2 (the diagonals)
Square	4
Regular $n$-gon	$n$
Circle	Infinitely many

Rotational Symmetry

A figure has rotational symmetry of order $n$ if it maps onto itself after rotations of $\frac{360°}{n}$.

• Order 1: only 360° maps it to itself — no meaningful rotational symmetry
• Order 2: maps to itself at 180° and 360°
• Order $n$: maps to itself at $\frac{360°}{n}$, $\frac{2 \times 360°}{n}$, ..., $360°$

The angle of rotation for the smallest rotation is $\frac{360°}{n}$.

Rotational Symmetry for Common Figures

Figure	Rotational symmetry order	Smallest angle
Scalene triangle	1 (none)	—
Isosceles triangle	1 (none)	—
Equilateral triangle	3	120°