Radius, diameter, circumference and the meaning of pi | Year 7

01 / LEARN

Understand the idea.

Three different lengths

A radius runs from the centre to the circle. A diameter runs from one side to the other through the centre, so it consists of two radii. The circumference is the length all the way around the curved boundary.

All three measurements are lengths. Before calculating, check the endpoints of the labelled segment: a line starting at the centre gives a radius, while one passing through the centre from boundary to boundary gives a diameter.

d = 2 r

Why pi appears

For every circle, the ratio of circumference to diameter is the constant $π$ . This gives $C = π d$ , or $C = 2 π r$ because $d = 2 r$ . Multiplying by $π$ scales a straight diameter length to the curved boundary length.

Leave $π$ in an exact answer. A decimal such as $3.14$ is only an approximation to this irrational number. If a decimal circumference is requested, retain full precision during the calculation and round at the end.

C = π d = 2 π r

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Radius supplied

A circle has radius 5 cm. Find its circumference exactly.

Worked solution

1
Convert the supplied radius to diameter.
$d = 2 r = 2 \times 5 = 10 cm$
A diameter consists of two radii.
2
Multiply the diameter by pi.
$C = π d = π \times 10$
Circumference divided by diameter is pi for every circle.
3
Keep the exact form.
$C = 10 π cm$
Replacing pi with a decimal would make the answer approximate.

Answer

$10 π cm$

EXAMPLE 02

Working backwards

A circle has circumference 18π cm. Find its radius.

Worked solution

1
Use the circumference-diameter relationship.
$18 π = π d$
Both circumference and diameter are expressed in centimetres.
2
Divide both sides by pi.
$d = 18 cm$
Pi is a nonzero common factor.
3
Halve the diameter.
$r = \frac{18}{2} = 9 cm$
The radius is half the diameter.

Answer

9 cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

A circle has diameter 14 cm. Give its radius and exact circumference.

Show answer & reasoning

$7 cm$ and $14 π cm$

Half of the $14 cm$ diameter is the radius, $7 cm$ . Multiplying the diameter by $π$ gives the exact circumference $14 π cm$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

A circle has radius 7 cm. Find its diameter.
1 mark
A circle has diameter 9 m. Find its radius.
1 mark
Find the exact circumference of a circle of radius 4 cm.
2 marks
A wheel has diameter 20 cm. Using $π \approx 3.14$ , find the distance travelled in one complete turn without slipping.
2 marks
A circle has circumference $24 π$ cm. Find its diameter and radius.
2 marks
A semicircle has diameter 10 cm. Find its complete perimeter, including the straight edge, exactly.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A circle has radius 7 cm. Find its diameter.
1 mark
$2 \times 7 = 14$ cm.
14 cm
A circle has diameter 9 m. Find its radius.
1 mark
$9 \div 2 = 4.5$ m.
4.5 m
Find the exact circumference of a circle of radius 4 cm.
2 marks
$Diameter = 8$ cm.
$C = π \times 8$ .
$8 π cm$
A wheel has diameter 20 cm. Using $π \approx 3.14$ , find the distance travelled in one complete turn without slipping.
2 marks
One turn travels one circumference.
$3.14 \times 20 = 62.8$ cm.
62.8 cm
A circle has circumference $24 π$ cm. Find its diameter and radius.
2 marks
Divide $C$ by $π$ to find $d = 24$ cm.
Halve the diameter to find $r = 12$ cm.
Diameter 24 cm; radius 12 cm
A semicircle has diameter 10 cm. Find its complete perimeter, including the straight edge, exactly.
3 marks
Full circumference would be $10 π$ cm.
The curved half is $5 π$ cm.
Add the straight diameter of 10 cm.
$(5 π + 10) cm$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 7 measurement and spatial representation: areas of triangles/parallelograms, prism volumes, circle circumference, nets and views. Precision is integrated into practical measurement; circle area is not introduced here.

Official curriculum source ↗

In this lesson

Understand the idea.

Three different lengths

Why pi appears

See the method in action.

Radius supplied

Convert the supplied radius to diameter.

Multiply the diameter by pi.

Keep the exact form.

Working backwards

Use the circumference-diameter relationship.

Divide both sides by pi.

Halve the diameter.

Try it yourself.

Check your understanding

Keep practising