KS3 Maths · Year 7 · Chapter 14: Circles · Lesson 1

The circumference of a circle

Year 7About 70 minutesCalculator allowed47 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $21.7 \div 7$ .
[1 mark]
Answer
A2
Needed today
A circle has a radius of $4.5$ cm. Write down its diameter.
[1 mark]
Answer
A3
Last lesson
$y$ is inversely proportional to $x$ . When $x = 4$ , $y = 9$ . Find $y$ when $x = 6$ .
[1 mark]
Answer
A4
From chapter 11
A map has a scale of $1 : 50000$ . How many kilometres does $4$ cm on the map stand for?
[1 mark]
Answer

Example, then your turn

Year 7

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

Example 1

$O$ is the centre of the circle. Write down the name of each line $P$ , $Q$ , $R$ and $S$ .

Hint Does the line go through

O

? Does it end on the circle? Does it only touch the circle?

B1
$O$ is the centre. Write down the name of each part $W$ , $X$ , $Y$ and $Z$ . $W$ and $X$ are the shaded regions, $Y$ is the thick curve and $Z$ is the line.
[4 marks]

Example 2

Sam measured three circles, in cm. Work out $C \div d$ for each one, to $1$ decimal place. What do you notice?

Hint Divide each circumference by its diameter, then round.

B2
The lengths are in cm. Work out $C \div d$ for each circle, to $1$ decimal place. Then copy and complete: "The circumference is about $\dots$ times the diameter."
[3 marks]
Answer $\dots$ , $\dots$ , $\dots$ ; about $\dots$ times

Example 3

Use $C \approx 3.1 \times d$ to estimate the circumference of a circle with a radius of $6$ cm.

Hint The rule uses the diameter. Double the radius first.

B3
Use $C \approx 3.1 \times d$ to estimate the circumference of a circle with a radius of $8.5$ cm.
[2 marks]
Answer $\dots$ cm

Example 4

A circular pond has a circumference of $62$ m. Use $C \approx 3.1 \times d$ to estimate its diameter and its radius.

Hint Work backwards: divide the circumference by

3.1

B4
A hoop has a circumference of $155$ cm. Use $C \approx 3.1 \times d$ to estimate its diameter and its radius.
[3 marks]
Answerdiameter $\dots$ cm, radius $\dots$ cm

Example 5

The circle has diameter $10$ cm. The sides of the regular hexagon equal the radius. The sides of the square equal the diameter. Find each perimeter. What does this tell you about the circumference?

Hint The radius is half the diameter. The hexagon has

6

sides and the square has

4

B5
The same shapes, but the diameter is $7$ cm. Find each perimeter. The circumference is between which two lengths?
[3 marks]
Answerbetween $\dots$ cm and $\dots$ cm

Practice

Year 7

C1
$O$ is the centre of the circle. $A$ and $B$ are on the circle. $O A = 5$ cm and the perimeter of triangle $O A B$ is $16$ cm. Work out the length of the chord $A B$ . Give a reason.
[2 marks]
Not drawn accurately
C2
Jo measured four circles, in cm. One circumference was measured wrongly. Which one? Give a reason.
[2 marks]

C3
Write true or false for each one. A: A diameter is also a chord. B: A tangent touches a circle at two points. C: The radius is twice the diameter.
[3 marks]
AnswerA $\dots$ B $\dots$ C $\dots$
C4
A circle has a radius of $6.5$ cm. What is the length of the longest chord you can draw in it? Give a reason.
[2 marks]

Exam-style questions

GCSE Foundation

D1
The wheel of a bicycle has a diameter of $64$ cm. Use circumference $\approx 3.1 \times$ diameter.
(a)
Estimate how far the bicycle moves when the wheel turns once.
[2 marks]
Answer $\dots$ cm
(b)
The bicycle travels $50$ m. Estimate the number of complete turns the wheel makes. You must show your working.
[3 marks]
Answer
Total for D1: 5 marks

D2
Amira says, "The circumference of a circle is about $2$ times its diameter, because the diameter goes across and back."
A circle has a diameter of $9$ cm and a circumference of $28.3$ cm. Use these to show that Amira is wrong.
[2 marks]
D3
A window is a rectangle with a semicircle on top. The rectangle is $1.2$ m wide and $1.5$ m tall. Estimate the total length of the frame round the outside of the window. Use circumference $\approx 3.1 \times$ diameter. You must show your working.
[4 marks]
Not drawn accurately
Answer $\dots$ m
D4
A circular table has a circumference of $4.65$ m. Use circumference $\approx 3.1 \times$ diameter.
(a)
Estimate the diameter of the table.
[2 marks]
Answer $\dots$ m
(b)
A round cloth covers the table and hangs $0.25$ m over the edge all the way round. Estimate the circumference of the cloth.
[3 marks]
Answer $\dots$ m
Total for D4: 5 marks

Extension

Stretch

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

E1
A rope fits tightly round a tree trunk with a circular cross-section of diameter $0.8$ m. Another rope fits tightly round a circular tower of diameter $20$ m. Each rope is made longer so that it sits $1$ m away from the surface all the way round. Use circumference $\approx 3.1 \times$ diameter. How much longer must each rope be? What do you notice?
[3 marks]
Hint 1 · What to try
Work out the new diameter for the tree. The rope is $1$ m away on both sides.
Hint 2 · The first line
Tree: the diameter goes from $0.8$ m to $2.8$ m. Work out both circumferences.
Hint 3 · The full method
Tree: $3.1 \times 2.8 - 3.1 \times 0.8 = 8.68 - 2.48 = 6.2$ m. Tower: $3.1 \times 22 - 3.1 \times 20 = 68.2 - 62 = 6.2$ m. Both are $6.2$ m longer: the extra is always $3.1 \times 2$ .
Answer $\dots$ m and $\dots$ m

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can name the parts of a circle: radius, diameter, chord, tangent, arc, sector and segment.
I know that the circumference is always about $3.1$ times the diameter.
I can estimate a circumference from a radius or diameter, and a diameter from a circumference.

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