Resources For Teachers For Tutors For Students & Parents AI New Pricing
Create Free Account
Year 6 Maths 2D Shapes, Position & Transformations

Tessellations

20 practice questions 0 video lessons Theory + worked examples
Practice 20 questions
Practice questions

Every question with a fully worked solution.

Start practising
Create a free accountTrack your progress and save your work as you go.
Create free account

Theory

A tessellation is a pattern of shapes that covers a surface with no gaps and no overlaps, like floor tiles or a honeycomb. Shapes fit together at a point only if their angles there add to exactly \(360^\circ\). This Year 6 maths page shows why the only regular shapes that tessellate on their own are equilateral triangles, squares and regular hexagons, and how other shapes can tessellate together.

A tessellation (tiling) repeats shapes to cover a flat surface with no gaps and no overlaps.

A regular polygon has all sides equal and all angles equal. Each interior angle is the angle sum divided by the number of angles: a regular decagon has \(1440^\circ \div 10 = 144^\circ\).

Where corners meet at a point, the angles must add to \(360^\circ\). So a regular shape tessellates on its own only if its interior angle divides exactly into \(360^\circ\).

Some shapes that do not tessellate alone can tessellate together, such as regular octagons with squares.

Honeycomb tessellation of hexagons Regular hexagons in three colours covering a rectangle with no gaps or overlaps; three hexagons meet at every corner.
Regular hexagons: \(3 \times 120^\circ = 360^\circ\) at every corner.
Two regular nonagons at a point Two regular nonagons placed round a point leave a red wedge-shaped gap of 80 degrees. 80°
Nonagons: \(2 \times 140^\circ = 280^\circ\), leaving an \(80^\circ\) gap. They do not tessellate alone.
Octagons and squares tessellation Regular octagons with small squares filling the gaps between them, covering a rectangle with no gaps.
Regular octagons alone leave square-shaped gaps; squares fill them exactly.

Regular shapes at a point (the triangle is equilateral):

ShapeAngleAt a pointAlone?
Triangle\(60^\circ\)\(6\)Yes
Square\(90^\circ\)\(4\)Yes
Nonagon\(140^\circ\)—No
Hexagon\(120^\circ\)\(3\)Yes
Decagon\(144^\circ\)—No
\[ 360^\circ \div \text{angle} = \text{whole number} \]
360÷ angle = whole number
Only three regular shapes. Equilateral triangles, squares and regular hexagons are the only regular shapes that tessellate on their own.
Any triangle, any quadrilateral. Every triangle and every four-sided shape can tessellate, even if it is not regular.

Does a regular shape tessellate?

  1. Find the interior angle: angle sum \(\div\) number of angles.
  2. Divide \(360^\circ\) by it.
  3. Whole number? It tessellates. Otherwise there is a gap or an overlap.

Do two shapes tessellate together?

  1. Add the angles of the shapes that meet at one point.
  2. Exactly \(360^\circ\)? They can fit together there.
Example 1 — Hexagons
Each angle of a regular hexagon is \(120^\circ\). Show that regular hexagons tessellate.
Solution
\(3 \times 120\)\(=\)\(360\)
3×120=360

Exactly \(3\) angles of \(120^\circ\) fit round a point with no gap.

Example 2 — Decagons
A regular decagon has \(10\) equal angles adding to \(1440^\circ\). Does it tessellate on its own?
Solution
\(1440 \div 10\)\(=\)\(144\)
\(2 \times 144\)\(=\)\(288\)
\(3 \times 144\)\(=\)\(432\)
144288 is too few ,432 too many

No. Two leave a gap and three overlap.

Example 3 — Nonagons
A regular nonagon has \(9\) equal angles adding to \(1260^\circ\). Two nonagons meet at a point. How big is the gap?
Solution
\(1260 \div 9\)\(=\)\(140\)
\(360 - 2 \times 140\)\(=\)\(80\)
360−280=80

\(80^\circ\). No regular polygon has an \(80^\circ\) angle, so it cannot be filled with a regular shape.

Example 4 — Dodecagons and triangles
A regular dodecagon (\(12\) sides) has angles of \(150^\circ\). Can two dodecagons and an equilateral triangle meet at a point?
Solution
\(150 + 150 + 60\)\(=\)\(360\)
150+150+60=360

Yes: the angles make exactly \(360^\circ\).

Example 5 — Squares and triangles
Can \(3\) equilateral triangles and \(2\) squares meet at a point?
Solution
\(3 \times 60 + 2 \times 90\)\(=\)\(360\)
180+180=360

Yes: \(180^\circ + 180^\circ = 360^\circ\).

Example 6 — Any triangle
A triangle has angles \(50^\circ\), \(60^\circ\) and \(70^\circ\). Explain how copies of it can tessellate.
Solution
\(2 \times (50 + 60 + 70)\)\(=\)\(360\)
2×180=360

Turn copies so each angle appears twice at a point: \(2 \times 180^\circ = 360^\circ\).

Example 7 — Hexagons and triangles
Can \(2\) regular hexagons and \(2\) equilateral triangles meet at a point?
Solution
\(2 \times 120 + 2 \times 60\)\(=\)\(360\)
240+120=360

Yes: \(240^\circ + 120^\circ = 360^\circ\).

Example 8 — Find the mistake
Ivy says only regular shapes can tessellate. Is she right?
Solution

No. Rectangles (bricks) tessellate, and so does any triangle or any four-sided shape, even though they are not regular.

Common pitfalls

Gaps or overlaps. A tessellation has neither.
Thinking more sides always works. Regular octagons and decagons do not tessellate alone.
Using the wrong total. Angles meeting at a point must add to \(360^\circ\).
Thinking shapes must be regular. Rectangles and any triangles tessellate too.

Frequently asked questions

What is a tessellation?

A pattern of shapes covering a surface with no gaps and no overlaps.

Which regular shapes tessellate on their own?

Equilateral triangles, squares and regular hexagons.

Why do regular hexagons tessellate?

Each angle is \(120^\circ\) and \(3 \times 120^\circ = 360^\circ\).

Why do regular decagons not tessellate?

Each angle is \(144^\circ\), and \(360 \div 144\) is not a whole number: two leave a gap and three overlap.

Can two different shapes tessellate together?

Yes, if their angles at each point add to 360°, like octagons and squares.

More in 2D Shapes, Position & Transformations
Plot & locate points on the coordinate plane (four quadrants)
20 questions
Translate, reflect & rotate shapes
20 questions