Tessellations | Year 6 | Ascent Library

01 / LEARN

Understand the idea.

Cover a surface

A tessellation covers a flat surface without gaps or overlaps and can continue beyond the patch drawn. Repeating colours or shapes alone is not enough: the tiles must meet so that every part of the surface is covered once.

Rectangles tessellate in rows because their edges line up and the next row fits against the previous one. Slides, turns or flips can help tiles fit. The colours highlight repetition, but the fit depends on the shapes and their angles.

Check where corners meet

At an interior meeting point, tile corners must fill one whole turn of $360^{\circ}$ . Four square corners give $4 \times 90^{\circ} = 360^{\circ}$ ; six equilateral-triangle corners give $6 \times 60^{\circ} = 360^{\circ}$ . These are useful checks where corners meet.

A $360^{\circ}$ corner total checks one meeting point; the edges must also fit as the pattern continues. Three regular-pentagon corners total only $324^{\circ}$ , leaving a gap, so that particular arrangement cannot cover the surface completely.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Squares meeting

A square has 90° corners. How many square corners fill one point in a square tiling?

Worked solution

1
Identify the angle to be filled.
$full turn = 360^{\circ}$
An interior point in a square tiling must be surrounded with no uncovered angle.
2
Divide by the size of one square corner.
$360^{\circ} \div 90^{\circ} = 4$
Each square contributes one 90-degree corner at that meeting point.
3
Check the fit.
$4 \times 90^{\circ} = 360^{\circ}$
Four square corners exactly fill the turn without an angle gap or overlap.

Answer

Four square corners.

EXAMPLE 02

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.

Worked solution

1
Add the three pentagon corners.
$3 \times 108^{\circ} = 324^{\circ}$
Each of the three regular pentagons contributes a 108-degree corner.
2
Find the uncovered angle.
$360^{\circ} - 324^{\circ} = 36^{\circ}$
The full turn is larger than the total angle covered by the three corners.
3
Judge this arrangement.
$36^{\circ} > 0^{\circ}$
The remaining gap violates the no-gaps requirement, so this arrangement is not a tessellation.

Answer

A 36° gap remains, so that arrangement does not tessellate.

03 / YOUR TURN

Try it yourself.

Check your understanding

Three regular hexagon corners are each 120°. Do they fill one turn?

Show answer & reasoning

Yes.

Three corners of $120^{\circ}$ total $3 \times 120^{\circ} = 360^{\circ}$ , so they fill one turn exactly. This checks the angles at that meeting point.

Keep practising

6 worksheet questions · 10 marks

Open worksheet Worked solutions

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 $60^{\circ}$ . How many meet to fill $360^{\circ}$ ?
1 mark
Each square corner is $90^{\circ}$ . Show why four fit around a point.
1 mark
Each regular hexagon corner is $120^{\circ}$ . How many must meet at a point?
1 mark
Three regular pentagon corners of $108^{\circ}$ 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 $60^{\circ}$ . How many meet to fill $360^{\circ}$ ?
1 mark
$360^{\circ} \div 60^{\circ} = 6$ .
6
Each square corner is $90^{\circ}$ . Show why four fit around a point.
1 mark
Their corners fill one full turn.
$4 \times 90^{\circ} = 360^{\circ}$ .
Each regular hexagon corner is $120^{\circ}$ . How many must meet at a point?
1 mark
$360^{\circ} \div 120^{\circ} = 3$ .
3
Three regular pentagon corners of $108^{\circ}$ meet. Calculate the gap.
2 marks
$3 \times 108^{\circ} = 324^{\circ}$ .
$360^{\circ} - 324^{\circ} = 36^{\circ}$ .
$36^{\circ}$
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.

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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 ↗

In this lesson

Understand the idea.

Cover a surface

Check where corners meet

See the method in action.

Squares meeting

Identify the angle to be filled.

Divide by the size of one square corner.

Check the fit.

Why regular pentagons leave a gap

Add the three pentagon corners.

Find the uncovered angle.

Judge this arrangement.

Try it yourself.

Check your understanding

Keep practising