KS3 Maths · Year 8 · Chapter 3: Polygons · Lesson 3

Tessellations and regular polygons

Year 8About 60 minutesNo calculator45 marks

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Date

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Quick questions to start.

A1
Last lesson
Work out the size of each interior angle of a regular octagon.
[1 mark]
Answer
A2
Needed today
Work out $360 \div 72$ .
[1 mark]
Answer
A3
Year 7
Three angles meet at a point. Two of them are $12 5^{\circ}$ and $14 0^{\circ}$ . Work out the third angle.
[1 mark]
Answer
A4
From chapter 2
Factorise $6 x + 15$ .
[1 mark]
Answer

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Explain why regular hexagons tessellate.

Not drawn accurately

Hint Shapes tessellate when copies fit round a point with no gap. Work out the interior angle first.

B1
Does a regular polygon with $18$ sides tessellate? Give a reason for your answer.
[2 marks]

Example 2

Two regular decagons are placed side by side at a point. Work out the size of the gap left at the point.

Hint Find the interior angle of a regular decagon. The angles at a point add up to

36 0^{\circ}

B2
Two regular dodecagons ( $12$ sides) are placed side by side at a point. Work out the size of the gap. Which regular polygon would fill the gap exactly?
[3 marks]
Answergap $\dots \dots^{\circ}$ polygon $\dots\dots\dots\dots$

Example 3

Show that a square and two regular octagons fit together exactly at a point.

Not drawn accurately

Hint Work out the interior angle of each shape. Do they add up to

36 0^{\circ}

B3
Show that a square, a regular hexagon and a regular dodecagon fit together exactly at a point.
[3 marks]

Example 4

Two regular pentagons and one other regular polygon fit together exactly at a point. How many sides does the other polygon have?

Hint Find the angle that is left. Then find the exterior angle of the polygon that has it.

B4
A square, a regular pentagon and one other regular polygon fit together exactly at a point. How many sides does the other polygon have?
[3 marks]
Answer $\dots\dots$ sides

Year 8

C1
Three regular pentagons are placed side by side at a point. Work out the size of the gap.
[2 marks]
Answer
C2
Explain why copies of any quadrilateral can tessellate.
[2 marks]

C3
Only three regular polygons tessellate on their own. Name them. Use the fact that the smallest interior angle of a regular polygon is $6 0^{\circ}$ .
[2 marks]
Answer
C4
Two regular hexagons and some equilateral triangles fit together exactly at a point. How many triangles are there?
[2 marks]
Answer $\dots\dots$ triangles
C5
A square and an equilateral triangle are placed side by side at a point. Could one regular polygon fill the gap? Give a reason.
[2 marks]

GCSE Foundation

D1
Ben says, "Every regular polygon with an even number of sides tessellates."
Show that Ben is wrong.
[3 marks]

D2
Equilateral triangles and squares fit together exactly at a point. There is at least one of each shape. How many triangles and how many squares are there? You must show your working.
[4 marks]
Answer $\dots\dots$ triangles $\dots\dots$ squares
D3
Two regular pentagons and a rhombus $P QR S$ fit together exactly at the point $P$ .
Not drawn accurately
(a)
Work out the size of each interior angle of a regular pentagon.
[2 marks]
Answer
(b)
Work out the size of angle $QP S$ .
[1 mark]
Answerangle $QP S = \dots \dots^{\circ}$
(c)
Work out the size of angle $P QR$ . Give a reason.
[2 marks]
Answerangle $P QR = \dots \dots^{\circ}$
Total for D3: 5 marks
D4
In a regular polygon, the interior angle is $4$ times the exterior angle. Does the polygon tessellate? You must show your working.
[4 marks]

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can explain why a regular polygon does or does not tessellate.
I can work out the gap left when regular polygons meet at a point.
I can find mixes of regular polygons that fit together at a point.

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