The idea, step by step
Cover a surface
A tessellation is a pattern of shapes covering a flat surface without gaps or overlaps. A small drawn patch shows how the pattern can continue beyond its boundary. Tiles may be translated, reflected or rotated to fit.
Rectangles make a simple tessellation: matching edges meet and the next row covers the same width. Colour can help show the repeating unit, but it is the shapes fitting together that matters.
Check where corners meet
At an interior meeting point, the angles of the tiles must fill 360°. Four square corners each contribute 90°, so they fit exactly. Six 60° corners of equilateral triangles also fill a turn.
Matching a 360° total is a useful local check; edges must also join to continue the pattern. A few shapes that almost meet are not enough. For example, three regular pentagon corners of 108° leave a 36° gap.
A common trap
Calling any repeated design a tessellation.
Instead: Check that the shapes cover the surface continuously, with no gaps and no overlaps.
See the thinking
Squares meeting
A square has 90° corners. How many square corners fill one point in a square tiling?
- Angles around a point total 360°.
- 360 ÷ 90 = 4.
Why regular pentagons leave a gap
A regular pentagon corner is 108°. Three such corners meet at a point. Find the gap and explain why this arrangement is not a tessellation.
- Three corners total 3 × 108° = 324°.
- 360° − 324° = 36°.
- The uncovered gap means the surface is not completely covered.
Check your understanding
Three regular hexagon corners are each 120°. Do they fill one turn?
Show answer
Yes.
3 × 120° = 360°.
Your turn
6 questions · 10 marks · Work at your own pace.
Read the questions online
- State the two conditions a repeating shape pattern must meet to be a tessellation.2 marks
- Each equilateral triangle corner is 60°. How many meet to fill 360°?1 mark
- Each square corner is 90°. Show why four fit around a point.1 mark
- Each regular hexagon corner is 120°. How many must meet at a point?1 mark
- Three regular pentagon corners of 108° meet. Calculate the gap.2 marks
- Draw a repeating tessellation of rectangles in the space. Show at least two rows and three columns. Explain how it continues.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- State the two conditions a repeating shape pattern must meet to be a tessellation.2 marks
Every part of the covered surface belongs to one tile, not zero tiles or two tiles.
No gaps and no overlaps. - Each equilateral triangle corner is 60°. How many meet to fill 360°?1 mark
360° ÷ 60° = 6.
6 - Each square corner is 90°. Show why four fit around a point.1 mark
Their corners fill one full turn.
4 × 90° = 360°. - Each regular hexagon corner is 120°. How many must meet at a point?1 mark
360° ÷ 120° = 3.
3 - Three regular pentagon corners of 108° meet. Calculate the gap.2 marks
3 × 108° = 324°.
360° − 324° = 36°.
36° - Draw a repeating tessellation of rectangles in the space. Show at least two rows and three columns. Explain how it continues.3 marks
Repeat the same rectangle along a row.
Stack matching rows.
Continue in both directions.
Any correctly joined rectangular grid, at least 2 by 3, with matching edges and no gaps/overlaps.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Year 6 prism cross-sections, combined transformations and tessellations, with visual reasoning.
Official curriculum source ↗