KA-0085Line and rotational symmetry
4 points, difficulty 2 of 3Level 3–4about 60sHow many lines of symmetry does a regular hexagon have?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Two kinds of fold work here.
2The strategy
A regular polygon has folds through opposite corners and folds through the middles of opposite sides. Count each kind separately, then add — but count each line once, not once per end.
3The full solution
Three lines join opposite corners and three join the middles of opposite sides, giving 6 in all.
Solution
The reliable way
There are two families of fold. Three lines join opposite corners, and three lines join the midpoints of opposite sides. That is 6 lines of symmetry. Each line is one line, not two, even though it has two ends. The transferable idea: a regular polygon with n sides has exactly n lines of symmetry, and for an even n they split into two equal families.
The elegant way
A regular n-sided polygon has n lines of symmetry, so a hexagon has 6.
Why this is on the test: Counting only one family of folds is the standard undercount, and doubling for the two ends is the standard overcount.