Circumference and area of circles | Year 8

01 / LEARN

The idea, step by step

Two different measurements

Circumference measures the curved boundary. Area measures the region inside it. The diameter crosses the centre from one side to the other, so it is twice the radius. A radius goes from the centre to the boundary.

The ratio of circumference to diameter is the same for every circle: pi. Use $C = 2 π r$ or $C = π d$ . For area use $A = π r^{2}$ . Length units and squared units tell the reader which quantity you found.

Keep pi until the end

An exact answer such as $18 π$ cm is already a complete answer. When a decimal is requested, substitute the calculator’s pi value at the last step and round once. If a question supplies an approximation, use that approximation consistently.

Before calculating, write the radius explicitly. If the diameter is 10 cm, the radius is 5 cm. Squaring the diameter instead would make the calculated area four times too large.

C = 2 π r, A = π r^{2}

A common trap

Using $A = π d^{2}$ when the diameter is given.

Instead: The area formula uses radius. Halve the diameter before squaring.

02 / WORKED EXAMPLES

See the thinking

Example 1

Radius supplied

A circle has radius 3 cm. Find its circumference and area exactly.

Circumference: $C = 2 π (3) = 6 π$ cm.
Area: $A = π {(3)}^{2} = 9 π$ cm².
The first answer is a length; the second is an area.

Circumference

6 π

cm; area

9 π

cm².

Example 2

Diameter supplied

A circular mat has diameter 14 m. Find its area, leaving pi in the answer.

Halve the diameter: $r = 14 \div 2 = 7$ m.
Substitute the radius: $A = π {(7)}^{2} = 49 π$ m².
A 14 m square has area 196 m²; a circle inside that square must have smaller area.

49 π

m².

Check your understanding

A circle has radius 5 cm. What is its circumference?

Show answer

$10 π$ cm.

Use $2 π r = 2 π (5)$ . Do not square for circumference.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A circle has radius 2 cm. Find its diameter.
1 mark
Find the circumference of a circle with radius 6 cm.
2 marks
Find the area of a circle with radius 4 m.
2 marks
A circular sign has diameter 10 cm. Find its circumference and area.
3 marks
A radius doubles from 3 cm to 6 cm. By what factor does circumference change? By what factor does area change?
3 marks
A circular pond has diameter 8 m. A student writes its area as $64 π$ m². Explain and correct the error.
3 marks

Read the worked solutions

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

A circle has radius 2 cm. Find its diameter.
1 mark
$d = 2 r = 2 (2) = 4$ cm.
4 cm.
Find the circumference of a circle with radius 6 cm.
2 marks
$C = 2 π (6) = 12 π$ cm.
$12 π$ cm.
Find the area of a circle with radius 4 m.
2 marks
$A = π {(4)}^{2} = 16 π$ m².
$16 π$ m².
A circular sign has diameter 10 cm. Find its circumference and area.
3 marks
The radius is $10 / 2 = 5$ cm.
$C = π d = 10 π$ cm.
$A = π {(5)}^{2} = 25 π$ cm².
$10 π$ cm and $25 π$ cm².
A radius doubles from 3 cm to 6 cm. By what factor does circumference change? By what factor does area change?
3 marks
Circumference is proportional to $r$ , so its factor is 2.
Area is proportional to $r^{2}$ , so its factor is $2^{2} = 4$ .
Circumference doubles; area is multiplied by 4.
A circular pond has diameter 8 m. A student writes its area as $64 π$ m². Explain and correct the error.
3 marks
The radius is 4 m.
$A = π {(4)}^{2} = 16 π$ m².
They squared the diameter. Correct area: $16 π$ m².

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Circumference and area of circles; perimeter/area of composite shapes; choosing and converting area units; cross-sections of prisms; prism volume and capacity; mixed measurement applications.

Official curriculum source ↗

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