What is a tessellation?
A tessellation is a pattern made of one or more shapes that completely covers a plane with no gaps and no overlaps. Floor tiles and honeycombs are everyday examples.
Angles at a point
The key rule is that the angles meeting at any point (a vertex) must add up to exactly $360^\circ$. For a regular polygon to tessellate on its own, its interior angle must divide $360^\circ$ exactly.
Key formula
$\text{interior angle of a regular }n\text{-gon}=\frac{(n-2)\times 180^\circ}{n}$
Which regular polygons tessellate?
Only three regular polygons tessellate by themselves:
- Equilateral triangle: interior angle $60^\circ$, and $360^\circ\div 60^\circ=6$ meet at a point;
- Square: interior angle $90^\circ$, and $4$ meet at a point;
- Regular hexagon: interior angle $120^\circ$, and $3$ meet at a point.
Example
A regular pentagon has interior angle $\dfrac{(5-2)\times 180^\circ}{5}=108^\circ$. Since $360^\circ\div 108^\circ=3.33\ldots$ is not a whole number, it cannot tessellate alone.
Remember
Any triangle and any quadrilateral will tessellate; a regular polygon tessellates only when $360^\circ$ is divisible by its interior angle.