A line of symmetry is a line that cuts a shape into two halves that match exactly when folded along it. The fold test is not a metaphor. Cutting the shape out and folding it is the most reliable way for a young student to settle any disagreement.
How many each shape has
| Shape | Lines of symmetry |
|---|---|
| Square | 4, two through opposite sides and two through opposite corners |
| Rectangle that is not a square | 2, one horizontal and one vertical |
| Equilateral triangle | 3 |
| Isosceles triangle | 1 |
| Scalene triangle | 0 |
| Regular hexagon | 6 |
| Circle | infinitely many, since every line through the centre works |
| Parallelogram that is not a rhombus or rectangle | 0 |
A regular polygon has exactly as many lines of symmetry as it has sides. That is a quick check worth remembering.
The rectangle trap
Almost every student tries a diagonal of a rectangle and expects it to work. It does not. Fold a non square rectangle along its diagonal and the two triangles do not land on each other, because their sides are different lengths. A square is the special case where the diagonal fold does work.
Line symmetry is not the only kind
A shape has rotational symmetry if turning it less than a full circle leaves it looking unchanged. A parallelogram has no lines of symmetry at all, but turning it half a circle returns it to its starting appearance, so it does have rotational symmetry. The two ideas are separate and a shape can have either, both, or neither.
Common mistakes
Counting rectangle diagonals
Two lines for a rectangle, not four. Test with a fold whenever unsure.
Assuming letters are symmetric
Capital A, H, M, O, T, U, V, W, X and Y have vertical or horizontal symmetry. F, G, J, L, N, P, Q, R, S and Z have none. Guessing gives about half of them wrong.
Check yourself
How many lines of symmetry does a regular pentagon have?
Five. A regular polygon has one for each side.
Does the letter S have a line of symmetry?
No. It has rotational symmetry, since turning it half a circle looks the same, but no fold line works.
How many lines of symmetry does an isosceles triangle have?
One, running from the apex down through the middle of the unequal side.
Why does the diagonal of a rectangle fail the fold test?
Because it makes two triangles whose matching sides are different lengths, so folding leaves parts sticking out.
See also
Sources
- Line and rotational symmetry as taught in United States elementary geometry. Diagrams are drawn for this site.