Question
How many lines of symmetry does a regular hexagon have? Identify all of them.
Solution — Step by Step
A line of symmetry (or axis of symmetry) divides a figure into two congruent mirror images. If you fold the figure along this line, both halves coincide exactly. For a regular polygon, a line of symmetry must pass through the center of the figure.
A regular polygon has all sides equal and all angles equal. The number of lines of symmetry of a regular polygon equals the number of sides.
For a regular hexagon (6 sides): 6 lines of symmetry.
The 6 lines of symmetry fall into two categories:
Type 1 — Through opposite vertices (3 lines): Label the vertices 1, 2, 3, 4, 5, 6. Lines passing through pairs of opposite vertices:
- Vertex 1 to Vertex 4
- Vertex 2 to Vertex 5
- Vertex 3 to Vertex 6
Each of these lines bisects two opposite angles and passes through two vertices.
Type 2 — Through midpoints of opposite sides (3 lines): Lines passing through the midpoints of opposite sides:
- Midpoint of side 1-2 to midpoint of side 4-5
- Midpoint of side 2-3 to midpoint of side 5-6
- Midpoint of side 3-4 to midpoint of side 6-1
These lines are perpendicular bisectors of the sides and pass through no vertices.
For regular polygons with an even number of sides ():
- lines through pairs of opposite vertices
- lines through midpoints of opposite sides
- Total: lines
For a hexagon (): lines ✓
For regular polygons with an odd number of sides ():
- All lines connect each vertex to the midpoint of the opposite side
For an equilateral triangle (): 3 lines, each from a vertex to the midpoint of the opposite side.
Why This Works
A regular hexagon has the highest degree of symmetry of any regular polygon commonly studied in school. The 6-fold rotational symmetry (, , , , , rotations) corresponds to exactly 6 lines of reflective symmetry — these are related by the symmetry group of the hexagon (the dihedral group ).
Each line of symmetry is also an axis about which the hexagon can be reflected, mapping it onto itself. This is why regular hexagons tile the plane so perfectly — think of honeycomb structures.
Alternative Method
You can also count by drawing: place a regular hexagon with one vertex at the top. Draw all possible lines through the center. You’ll find exactly 6 such lines that produce mirror images — no more, no fewer.
Quick rule to memorise: “A regular n-gon has exactly n lines of symmetry.” This works for all regular polygons — equilateral triangle (3), square (4), regular pentagon (5), regular hexagon (6), and so on.
Common Mistake
Students sometimes count only the lines through opposite vertices (3 lines) and miss the 3 lines through midpoints of opposite sides. The correct answer is 6, not 3. Always check for both types of symmetry lines in even-sided regular polygons.