Symmetry · Usually met around P3
How many lines of symmetry a shape has
Try every fold: upright, flat, slanting. A circle has infinitely many.
How to do it
Method
Try every fold: upright, flat, slanting. A circle has infinitely many
Rectangle 2, square 4, regular hexagon 6
Another way
Corners and middles of sides
Regular hexagon: 3 through corners + 3 through middles of sides = 6
In full
Try each fold and count the ones where the halves match exactly: the upright middle, the flat middle, the two diagonals, and slanting lines through a corner or the middle of a side. A rectangle has just 2, upright and flat; folded on a diagonal its corners stick out. A square has 4; an equilateral triangle 3; a regular polygon has as many lines of symmetry as it has sides. A circle matches folded along any straight line through its centre (a diameter), so it has as many lines of symmetry as you like: infinitely many. Another way: when a regular polygon has an odd number of sides, each line goes through a corner and the middle of the opposite side; when it has an even number, half go through two opposite corners and half through the middles of two opposite sides: a regular hexagon has 3 + 3 = 6.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
How many lines of symmetry does this shape have?
Show the answer and working
Answer: 5 lines
Try every fold, upright, flat and slanting: 5 of them match exactly.
Example 2
How many lines of symmetry does this shape have?
- A. 4
- B. 1
- C. 2
- D. Infinitely many
Choose the right answer
Show the answer and working
Answer: B. 1
Try each fold and count; a circle matches on every diameter, so it has infinitely many.
Example 3
A regular polygon has 20 sides. Of its lines of symmetry, □ pass through two opposite corners and □ through the middles of two opposite sides.
Fill in each box
Show the answer and working
Answer: 10，10
20 is even: its 20 corners make 10 opposite pairs and its 20 sides 10 more, 20 lines in all.
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