Year 9 Maths The Thai Guy

Lesson 9.2 · 20 min

Tessellations

Which shapes fit together with no gaps or overlaps — and the angle rule that decides it.

In this lesson you will learn to…

  • Explain what a tessellation is
  • Use angles at a point to decide whether a regular polygon tessellates
  • Explain why every triangle and every quadrilateral tessellates
  • Find missing angles in a tessellation

Why this matters

Tessellations are everywhere — bathroom tiles, brick walls, honeycomb, paving, and the spectacular geometric patterns of Islamic art and Egyptian floor tiling. A designer needs to know in advance which shapes will cover a floor without gaps or awkward cutting.

The regular polygons

Regular polygonInterior angle360÷360 \div angleTessellates?
Triangle60∘60^\circ6✓
Square90∘90^\circ4✓
Pentagon108∘108^\circ3.33…✗
Hexagon120∘120^\circ3✓
Octagon135∘135^\circ2.67…✗

Bees build hexagonal cells for a reason: of the three shapes that tessellate, the hexagon encloses the most area for the least wall.

Mixing shapes

Two different regular polygons can tessellate together if their angles combine to 360∘360^\circ. Two octagons and a square: 135+135+90=360∘135 + 135 + 90 = 360^\circ — a common floor-tile pattern.

Irregular shapes too. Many irregular shapes tessellate — any triangle, any quadrilateral (even a concave one like an arrowhead), and some L-shapes and T-shapes. Try them on dotty paper.

Key rules and formulas

  • Tessellation

    A pattern of shapes that fit together with no gaps and no overlaps. Circles do not tessellate — they always leave gaps.

  • The angle test

    Where the corners meet, their angles must add up to exactly . For one regular polygon, its interior angle must divide exactly into 360.

  • Only three regular polygons

    Equilateral triangles (), squares () and regular hexagons (). A regular pentagon () does not: is not a whole number.

  • Every triangle and quadrilateral

    Rotate copies through about the midpoint of a side. Six copies of any triangle put all three angles in twice at one point (); four copies of any quadrilateral put all four angles in once ().
Try it

Does it tessellate?

5 sides
Name
pentagon
Interior angle sum = (n − 2) × 180°
(5 − 2) × 180° = 540°
Each interior angle
108°
Each exterior angle = 360° ÷ n
72°

Try 3, 4 and 6 sides: the interior angle divides exactly into . Try 5 or 8: it does not, so copies leave gaps.

Worked examples

Worked example 1

Regular pentagons

Explain why regular pentagons do not tessellate.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Angles in a triangle tessellation

In a tessellation of equilateral triangles, six angles to meet at one point. Find .

Try it yourself first, then reveal the solution one step at a time.

Practice: tessellations

8 questions

Eight questions. The key idea every time: do the angles at a point make exactly ?
  1. Question 1 of 8Choose all that applyFoundation
    Which of these regular polygons tessellate on their own?

    Select every correct answer — there may be more than one.

  2. Question 2 of 8Multiple choice
    Why is a pattern of circles not a tessellation?
  3. Question 3 of 8Step by step
    Test the regular pentagon.
    Does a regular pentagon tessellate?
    1. What is its interior angle?
      °

  4. Question 4 of 8Type the answer
    Six equal angles of equilateral triangles meet at a point in a tessellation. What is each angle?
    °
  5. Question 5 of 8True or false
    Every triangle, of any shape, will tessellate.
  6. Question 6 of 8Type the answer
    Four copies of a quadrilateral with angles , , and meet at a point in a tessellation, one of each angle. Find .
    °
  7. Question 7 of 8Choose all that applyChallenge
    Which combinations of regular polygons fit exactly round a point?

    Select every correct answer — there may be more than one.

  8. Question 8 of 8Fill the gaps
    Complete the definition.

    A tessellation is a pattern of shapes that fit together with no and no .