Symmetry (reflectional and rotational)
Reflectional and Rotational Symmetry
Reflectional and Rotational Symmetry is a topic in Transformations, Congruence & Proof in the California Common Core State Standards. It is aligned to Standard G-CO.3, which requires students to describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself.
A figure has symmetry when a reflection or a rotation maps it exactly onto itself.
Theory
A figure has symmetry when a transformation maps it exactly onto itself. Two kinds:
- Reflectional (line) symmetry: a line of symmetry acts as a mirror, mapping the figure onto itself.
- Rotational symmetry: a rotation about a center by less than a full turn maps the figure onto itself. The order is how many times this happens in \(360^\circ\).
Rotational symmetry of a regular polygon:
How to analyze symmetry
- Reflectional: look for mirror lines that fold the figure onto itself.
- Rotational: find the smallest turn (about the center) that maps it onto itself.
- Order \(=\dfrac{360^\circ}{\text{smallest angle}}\).
A square maps onto itself when reflected over each of these mirror lines: both diagonals and both midlines.
| \(\text{lines of symmetry}\) | \(=\) | \(4\) |
A regular \(n\)-gon has rotational symmetry of order \(n\).
| \(\text{order}\) | \(=\) | \(6\) |
Divide a full turn by the order \(n=5\).
| \(\dfrac{360^\circ}{5}\) | \(=\) | \(72^\circ\) |
The letter \(S\) looks the same after a \(180^\circ\) rotation but has no mirror line, so it has rotational symmetry (order 2) and no reflectional symmetry.
Common pitfalls
Frequently asked questions
What is a line of symmetry?
A line that acts as a mirror: reflecting the figure over it maps the figure exactly onto itself.
What is the order of rotational symmetry?
The number of times a figure maps onto itself during one full \(360^\circ\) turn about its center.
How do you find the angle of rotational symmetry?
Divide \(360^\circ\) by the order: a shape of order \(n\) has angle \(\dfrac{360^\circ}{n}\).
How many lines of symmetry does a regular polygon have?
A regular \(n\)-gon has \(n\) lines of symmetry and rotational symmetry of order \(n\).