A fold that matches
Fold a shape along a straight line. If the two halves land exactly on top of each other, with nothing sticking out, the fold line is a line of symmetry. The two halves are identical, matching halves: each one is the mirror image of the other.
That is why a line of symmetry is also called a mirror line. Stand a mirror on it, and the half in front of the mirror, together with its reflection, makes the whole shape again.
Fold this hexagon along the dashed line, and one half lands exactly on the other.
Some folds do not work
Not every line through the middle of a shape is a line of symmetry. This trapezium has exactly one: the line straight up and down through the middle of its top and bottom sides.
Fold it any other way and the halves do not match. Folded across, from top to bottom, the long bottom side lands on the short top side, and its ends stick out.
The trapezium's only line of symmetry runs straight up and down through its middle.
Count every fold
A square has four lines of symmetry. Two go through the middles of opposite sides, one up and down and one across. The other two are the diagonals, which join opposite corners. Folding along a diagonal lays one triangle exactly on the other, because all four sides of a square are the same length.
The four lines of symmetry of a square: two through the middles of opposite sides and two along the diagonals.
A rectangle has only two
A rectangle that is not a square has only two lines of symmetry: up and down through its middle, and across through its middle. Its diagonals are not lines of symmetry. Fold a rectangle along a diagonal and two corners stick out, because a long side lands on a short side.
A rectangle folds onto itself up and down and across, but not along its diagonals.
folded half ≠ other half: part of it sticks out past the edge, so this fold is not a line of symmetry
Find a fold where the two halves match
Folded corner to corner, the rectangle's half sticks out past its edges. Folded up and down or across, the halves match. Switch to the square, and its corner-to-corner fold matches too.
Regular shapes
A shape whose sides are all the same length and whose angles are all the same size is called a regular shape. A regular shape has one line of symmetry for every side: a regular pentagon has 5, and a regular hexagon has 6.
In the hexagon, 3 of the lines join opposite corners and 3 join the middles of opposite sides. Counting only the lines through corners gives 3, which misses half of them.
A regular hexagon has 6 lines of symmetry: 3 through opposite corners and 3 through the middles of opposite sides.
Worked example: Lines of Symmetry
Question How many lines of symmetry does a regular hexagon have? How many does a rectangle that is not a square have?
1.Regular hexagon: a line through two opposite corners folds it onto itself; there are 3 such lines.
Three lines through opposite corners. 2.A line through the midpoints of two opposite sides does too; there are 3 more.
Three more through the midpoints of opposite sides. 3.Total for the hexagon: 3 + 3 = 6 lines of symmetry.
The hexagon: 3 + 3 = 6. 4.Rectangle: the line through the midpoints of the two long sides, and the line through the midpoints of the two short sides: 2 lines.
The rectangle: only the two midpoint lines. 5.A diagonal of a rectangle is not a line of symmetry: folding along it sends a short side onto a long one.
A diagonal fold sends a short side onto a long one: not a mirror.
Answer: 6; 2
Common mistakes
- Counting a rectangle's diagonals: they look symmetric but a fold along one does not match the sides.
- Giving 3 for the hexagon by counting only the corner-to-corner lines.