A tessellation is a pattern of shapes that covers a flat surface with no gaps and no overlaps. Only three regular polygons can tessellate on their own — the equilateral triangle, the square, and the regular hexagon. Crucially, every triangle and every quadrilateral will always tessellate, making them especially important to understand at KS3.
What is a tessellation?
A tessellation (also called a tiling) is an arrangement of one or more shapes that completely covers a flat surface with no gaps and no overlaps. The shapes must fit together perfectly so that every part of the surface is covered and no shape crosses another.
You encounter tessellations every day: bathroom floor tiles, the honeycomb cells of a beehive, brick walls, and pavement stones are all real-life tessellations. In mathematics, we are more precise — we ask which shapes can tile an infinite flat plane under exactly these conditions.
Tessellations fall into three broad groups:
- Regular tessellations — made from a single type of regular polygon, repeated.
- Semi-regular tessellations — made from two or more types of regular polygon, with the same arrangement at every vertex.
- Irregular tessellations — made from non-regular shapes, such as any scalene triangle or general quadrilateral.
Which regular polygons can tessellate on their own?
Exactly three regular tessellations exist. The key to understanding why is the interior angle.
The interior angle of a regular polygon with n sides is:
Interior angle = (n − 2) × 180° ÷ n
For identical copies of a regular polygon to fit snugly around a single point, the angles at that point must add to exactly 360° — a full turn — with no gap and no overlap. This means the interior angle must divide 360° exactly (that is, 360 ÷ interior angle must be a whole number).
| Polygon | Sides | Interior angle | 360° ÷ angle | Tessellates alone? |
|---|---|---|---|---|
| Equilateral triangle | 3 | 60° | 6 — whole number ✓ | Yes — 6 meet at a point |
| Square | 4 | 90° | 4 — whole number ✓ | Yes — 4 meet at a point |
| Regular pentagon | 5 | 108° | 3.33… — not whole ✗ | No |
| Regular hexagon | 6 | 120° | 3 — whole number ✓ | Yes — 3 meet at a point |
| Regular heptagon | 7 | ≈128.6° | ≈2.8 — not whole ✗ | No |
| Regular octagon | 8 | 135° | 2.67 — not whole ✗ | No |
The only three regular tessellations are the triangular tiling, the square tiling, and the hexagonal tiling. A very common error is assuming that all regular polygons tessellate — they most definitely do not.
Why do triangles and quadrilaterals always tessellate?
This is one of the most satisfying results at KS3, and the reasoning is fully within reach of every student.
Any triangle tessellates. Take any triangle and rotate an identical copy through 180° about the midpoint of one of its edges. The two triangles join to form a parallelogram. Parallelograms can be translated (slid) to tile the whole plane without rotation. At each vertex in the final tiling, the angles from adjacent triangles always combine to give 360° — because you are fitting together complete sets of angles from two triangles, each summing to 180°, giving 360° in total at every meeting point.
Any quadrilateral tessellates. The four interior angles of any quadrilateral sum to 360°. Arrange four copies of the same quadrilateral around a single point, placing one of each angle at that point. The angles sum to exactly 360°, leaving no gap and no overlap. Translate this cluster in two directions to fill the entire plane.
This is why an exam question will never ask "does this particular triangle tessellate?" expecting the answer "no" — every triangle does, without exception.
What is the rule for a shape to tessellate?
The fundamental rule is:
The interior angles at every meeting point (vertex) in the tiling must sum to exactly 360°.
If angles sum to more than 360°, shapes overlap. If they sum to less than 360°, a gap is left. Either way, the tiling fails.
For a single regular polygon, this means checking whether 360° ÷ (interior angle) gives a whole number. Only 60°, 90°, and 120° satisfy this test — giving the equilateral triangle, square, and regular hexagon respectively.
For irregular shapes and semi-regular arrangements, the same vertex rule applies, but you can mix angles from different corners of the same shape, or from different shapes, as long as each meeting point in the finished tiling sums to exactly 360°.
What is a semi-regular tessellation?
A semi-regular tessellation (also called an Archimedean tessellation) uses two or more different types of regular polygon, arranged so that the exact same combination of polygons appears at every vertex throughout the tiling. Mathematicians have proved there are exactly eight semi-regular tessellations.
A well-known example is the 4.8.8 tessellation — one square and two regular octagons meet at every vertex. The angles are: 90° (square) + 135° (octagon) + 135° (octagon) = 360° ✓. This pattern appears on many decorative floor tiles and in some footballs.
Another familiar example is the 3.6.3.6 tessellation — alternating equilateral triangles and regular hexagons, with 60° + 120° + 60° + 120° = 360° ✓ at every vertex. The notation lists the number of sides of each polygon in order around any vertex.
Are there shapes that cannot tessellate?
Yes — most regular polygons cannot tile the plane on their own. Regular pentagons are the most important example for KS3.
Interior angle of a regular pentagon = (5 − 2) × 180° ÷ 5 = 540° ÷ 5 = 108°
Check: 360° ÷ 108° = 3.33… — not a whole number. Three pentagons around a point leave a gap of 360° − (3 × 108°) = 360° − 324° = 36°. Four would overlap by 432° − 360° = 72°. Neither arrangement works.
Once the interior angle of a regular polygon exceeds 120°, you cannot fit even three copies around a point without overlapping. This rules out regular heptagons (approximately 128.6°), octagons (135°), and all higher regular polygons from tessellating on their own.
It is worth knowing that certain irregular pentagons can tile the plane — there are fifteen known types — but for KS3 the key fact is simply: regular pentagons do not tessellate.
Frequently Asked Questions
Why do only three regular tessellations exist?
For a regular polygon to tessellate by itself, its interior angle must divide exactly into 360°, giving a whole number of copies around each vertex. Interior angles of regular polygons increase as the number of sides grows. Only 60° (equilateral triangle), 90° (square), and 120° (hexagon) divide 360° giving a whole number. Every regular polygon with more than six sides has an interior angle greater than 120°, making it impossible for even three copies to meet at a vertex without overlapping.
Can any irregular shape tessellate?
Not all irregular shapes tessellate, but the rule is unchanged: angles at every vertex in the tiling must sum to 360°. All triangles tessellate and all quadrilaterals tessellate — these two facts are essential at KS3. Some irregular pentagons also tessellate, but this is not guaranteed for all of them. As a general principle, the more sides a shape has, the harder it becomes to arrange copies without gaps.
What is the difference between a regular and a semi-regular tessellation?
A regular tessellation uses only one type of regular polygon throughout; a semi-regular tessellation uses two or more types but must have the identical vertex arrangement everywhere in the tiling. There are exactly three regular tessellations and eight semi-regular (Archimedean) tessellations. In both cases, the angles around each vertex still sum to exactly 360°.
How do I use the interior angle formula to check if a polygon tessellates?
Calculate the interior angle using the formula (n − 2) × 180° ÷ n, where n is the number of sides. Then divide 360° by that angle and check whether the result is a whole number. If it is, identical copies of that polygon can meet neatly at a point, and the polygon can tessellate on its own. If the result is not a whole number, the polygon cannot tessellate alone — though it may still appear in a semi-regular tessellation alongside other polygons.
Ready to work through tessellations step by step with a tutor who never just hands you the answer? Visit AI Tutors and ask Professor Pi — your KS3 maths companion.