What is symmetry?
Fold a picture of a butterfly along the middle. The two halves match exactly, one lying on top of the other. A figure like this is symmetrical, and the fold line is its line of symmetry (also called the axis of symmetry).
Symmetry is everywhere: leaves, faces, the Taj Mahal, rangoli designs, and the logos of many companies.
Line symmetry
A figure has line symmetry if a line divides it into two parts that are mirror images of each other. Every point on one side has a matching point on the other side, at the same distance from the line.
A simple test: trace the shape, fold the tracing along the line, and check whether the two halves fit exactly.
A figure may have:
no line of symmetry (a scalene triangle, the letter F),
one line of symmetry (an isosceles triangle, a kite, the letter A),
two or more lines of symmetry (a rectangle, a square, a circle).
Lines of symmetry of common shapes
Shape | Number of lines of symmetry |
|---|---|
Scalene triangle | 0 |
Isosceles triangle | 1 |
Equilateral triangle | 3 |
Rectangle (not a square) | 2 |
Square | 4 |
Rhombus (not a square) | 2 |
Parallelogram (not a rectangle or rhombus) | 0 |
Regular pentagon | 5 |
Regular hexagon | 6 |
Circle | infinitely many |
A regular polygon, with all sides equal and all angles equal, has as many lines of symmetry as it has sides.
A common mistake is to think a rectangle's diagonals are lines of symmetry. Fold a rectangle along a diagonal and the two halves do not fit. Only the lines joining the midpoints of opposite sides work.
Symmetry in letters
Capital letters of the English alphabet show different kinds of symmetry (in a simple block font):
Lines of symmetry | Letters |
|---|---|
Vertical line only | A, M, T, U, V, W, Y |
Horizontal line only | B, C, D, E, K |
Both vertical and horizontal | H, I, O, X |
None | F, G, J, L, N, P, Q, R, S, Z |
Mirror reflection
When you look in a mirror, your image is reversed left to right. If you raise your right hand, your image seems to raise its left hand.
The mirror acts as a line of symmetry between an object and its image:
the image is the same size and shape as the object,
each point of the image is as far behind the mirror as the matching point of the object is in front of it,
left and right are swapped.
That is why "AMBULANCE" is printed reversed on the front of an ambulance: drivers ahead read it the right way round in their rear-view mirrors.
Completing a symmetrical figure
To complete a figure given half of it and the line of symmetry, reflect each corner point across the line: measure its distance from the line and mark a point at the same distance on the other side. Then join the points in order.
Summary
A line of symmetry divides a figure into two mirror-image halves.
A regular polygon with n sides has n lines of symmetry; a circle has infinitely many.
A mirror image is the same size as the object, with left and right reversed.