Hexagon Tiling — Definition, Formula & Examples

Hexagon tiling is a pattern where regular hexagons fit together to cover a flat surface completely, leaving no gaps and no overlaps. It is one of only three regular polygon tilings possible.

A hexagonal tessellation is an edge-to-edge tiling of the Euclidean plane by congruent regular hexagons, where exactly three hexagons meet at every vertex and their interior angles sum to $360°$ .

Key Formula

\text{Interior angle} = \frac{(n-2) \times 180°}{n}

Where:

$n$ = Number of sides of the regular polygon (for a hexagon, n = 6)

How It Works

A regular hexagon has interior angles of

120°

each. At any vertex in the tiling, three hexagons meet, and

3 \times 120° = 360°

, which fills the space around that point perfectly. This is the key requirement for any regular polygon to tile the plane: copies of the polygon must fit around a single point with no angle left over. Only equilateral triangles (

60°

), squares (

90°

), and regular hexagons (

120°

) satisfy this condition among regular polygons.

Worked Example

Problem: Show that regular hexagons can tile the plane, but regular pentagons cannot.

Hexagon interior angle: Find the interior angle of a regular hexagon using n = 6.

\frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120°

Check hexagon vertex: See how many hexagons fit around one point: 360° ÷ 120° = 3, which is a whole number, so hexagons tile.

\frac{360°}{120°} = 3

Pentagon interior angle: Find the interior angle of a regular pentagon using n = 5.

\frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°

Check pentagon vertex: 360° ÷ 108° ≈ 3.33, which is not a whole number, so regular pentagons leave gaps and cannot tile the plane.

\frac{360°}{108°} \approx 3.33

Answer: Regular hexagons tile because exactly 3 fit around each vertex (3 × 120° = 360°). Regular pentagons do not tile because 360° is not evenly divisible by 108°.

Why It Matters

Hexagonal tilings appear in honeycombs, bathroom floors, and board games like Settlers of Catan. In engineering and materials science, hexagonal grids distribute stress evenly, which is why they show up in structures like graphene and satellite mirrors.

Common Mistakes

Mistake: Assuming any regular polygon can tile the plane if you use enough of them.

Correction: Only three regular polygons tile the plane on their own: equilateral triangles, squares, and regular hexagons. The interior angle must divide evenly into 360°.