Tessellations
Theory
A tessellation tiles a surface with copies of a shape and no gaps or overlaps. Shapes fit when the angles meeting at every point add to \(360^\circ\). Regular triangles, squares and hexagons tessellate on their own; regular pentagons do not.
A tessellation is a tiling that covers a flat surface with copies of a shape, leaving no gaps and no overlaps.
Copies fit together edge to edge. The key test is what happens at each meeting point: the angles that come together there must add to exactly \(360^\circ\), a full turn.
The interior angle of a regular shape with \(n\) sides is \(\dfrac{(n-2)\times 180^\circ}{n}\). A regular shape tessellates on its own only when that angle divides evenly into \(360^\circ\).
Three regular shapes do this: the equilateral triangle (\(6\times 60^\circ\)), the square (\(4\times 90^\circ\)) and the regular hexagon (\(3\times 120^\circ\)).
A single regular shape tessellates only when its interior angle divides evenly into \(360^\circ\). The table checks each one.
| Regular shape | Interior angle | \(360^\circ \div\) angle | Tessellates alone? |
|---|---|---|---|
| Triangle | \(60^\circ\) | \(6\) | Yes |
| Square | \(90^\circ\) | \(4\) | Yes |
| Pentagon | \(108^\circ\) | \(3\dfrac{1}{3}\) | No |
| Hexagon | \(120^\circ\) | \(3\) | Yes |
| Octagon | \(135^\circ\) | \(2\dfrac{2}{3}\) | No |
How to test whether a shape tessellates
- Find the interior angle. For a regular shape with \(n\) sides, work out \(\dfrac{(n-2)\times 180^\circ}{n}\).
- Divide \(360^\circ\) by that angle. This is how many copies would meet at one point.
- Check for a whole number. A whole number means the angles close the \(360^\circ\) exactly, so the shape tessellates. A remainder means a gap is left, so it does not.
Each interior angle is \(90^\circ\). Divide \(360^\circ\) by it.
| \(360^\circ \div 90^\circ\) | \(=\) | \(4\) |
4 squares fit exactly.
| \(\dfrac{(6-2)\times 180^\circ}{6}\) | \(=\) | \(\dfrac{720^\circ}{6}\) |
| \(=\) | \(120^\circ\) |
Each angle is \(120^\circ\).
| \(3\times 108^\circ\) | \(=\) | \(324^\circ\) |
| \(360^\circ - 324^\circ\) | \(=\) | \(36^\circ\) |
A \(36^\circ\) gap is left, so they do not tessellate.
| \(135^\circ + 135^\circ + 90^\circ\) | \(=\) | \(360^\circ\) |
Yes — together they tessellate.
Common pitfalls
Frequently asked questions
What is a tessellation?
A tessellation is a tiling that covers a flat surface with copies of a shape, leaving no gaps and no overlaps. The copies fit together edge to edge.
Why do the angles at a point have to add to 360 degrees?
There is a full turn of \(360^\circ\) around any point. If the tile angles meeting there add to exactly \(360^\circ\) they close the point with no gap and no overlap. Less leaves a gap; more forces an overlap.
Which regular shapes tessellate on their own?
Only the equilateral triangle, the square and the regular hexagon. Their interior angles \(60^\circ\), \(90^\circ\) and \(120^\circ\) each divide evenly into \(360^\circ\).
Why do regular pentagons not tessellate?
A regular pentagon's interior angle is \(108^\circ\), and \(360^\circ\) does not divide evenly by \(108^\circ\). Three pentagons cover \(324^\circ\) and leave a \(36^\circ\) gap, so they cannot tile a surface alone.
Can shapes that do not tessellate alone ever tile together?
Yes. Regular octagons leave square-shaped gaps, but a square fills each gap: \(135^\circ + 135^\circ + 90^\circ = 360^\circ\). Using more than one shape, many more patterns tessellate.