Symmetry
A shape has symmetry if a transformation maps it onto itself.
About this topic
A figure has line symmetry if a mirror line splits it into two matching halves, and rotational symmetry if it fits onto itself after turning by less than a full turn. The order of rotation is the number of times it fits in 360°. A regular n-gon has n lines of symmetry and rotational order n.
Symmetry is found in snowflakes, butterflies, logos, architecture and molecules, and it makes shapes balanced and easy to analyse.
Worked example: an equilateral triangle has 3 lines of symmetry and rotational order 3, since it fits onto itself after turns of 120°. A rectangle that is not a square has 2 lines of symmetry and rotational order 2. The letter H has both line and rotational symmetry, while the letter S has rotational symmetry only. A circle has infinite lines of symmetry.
Architects, jewellers and flag designers use symmetry for balance, and chemists use it to classify molecules. A common mistake is counting diagonal lines of symmetry for a rectangle; they do not work, since folding along a diagonal does not match the halves. Also, a parallelogram has rotational symmetry but no line symmetry. Test a line of symmetry by imagining a fold, and test rotation by tracing the shape and spinning the tracing about its centre.
Key ideas
- Line symmetry: mirror
- Rotational symmetry: turn
- A regular n-gon has n lines of symmetry
Quick check
Use cases: where Symmetry helps
- Homework: review the key idea behind Symmetry before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Symmetry without notes.
- Visual learning: play with the interactive lab to see how Symmetry changes when you adjust its inputs.
- Teaching a friend: explain Symmetry in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Symmetry?
A shape has symmetry if a transformation maps it onto itself.
What will I learn in Symmetry?
Line symmetry: mirror. Rotational symmetry: turn. A regular n-gon has n lines of symmetry.
Is Symmetry part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Symmetry lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Symmetry?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.