Area and Circumference of Circles (Year 8)

01 / LEARN

Understand the idea.

Two different measurements

A radius joins the centre to the circle's edge. A diameter crosses the centre and contains two radii, so $d = 2 r$ . First decide whether the question asks for the boundary length or the surface inside it.

Circumference is the distance once around the circle: $C = π d = 2 π r$ . Area counts the square units inside: $A = π r^{2}$ . Circumference uses length units; area uses squared units.

Keep pi until the end

Write the radius before substituting into a formula. A diameter of $10 cm$ means $r = 5 cm$ . Using the diameter as the radius doubles a circumference calculation and makes an area calculation four times as large.

Leaving $π$ in an answer keeps it exact: $18 π cm$ is a complete length. If a decimal is requested, use the calculator's $π$ value and round only the final result. Use a supplied approximation when instructed.

C = 2 π r, A = π r^{2}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Radius supplied

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

Worked solution

1
Identify the radius and the two formulas.
$r = 3 cm, C = 2 π r, A = π r^{2}$
The boundary and the enclosed region require different measurements.
2
Calculate the circumference.
$C = 2 π (3) = 6 π cm$
The circumference is twice the radius multiplied by pi.
3
Calculate the area.
$A = π {(3)}^{2} = 9 π {cm}^{2}$
Square the radius before multiplying by pi; area has square units.

Answer

Circumference $6 π$ cm; area $9 π$ cm².

EXAMPLE 02

Diameter supplied

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

A circle with its full diameter labelled 14 metres. The diameter passes through the centre; no area or radius answer is supplied.

Worked solution

1
Convert the diameter to a radius.
$r = \frac{14}{2} = 7 m$
A radius is half the diameter.
2
Substitute the radius into the area formula.
$A = π r^{2} = π {(7)}^{2}$
The area formula requires the radius, not the diameter.
3
Square the radius and retain pi.
$A = 49 π m^{2}$
Keeping pi gives the exact area requested.

Answer

$49 π$ m².

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$10 π$ cm.

Circumference uses $C = 2 π r$ . With $r = 5 cm$ , $C = 2 π (5) = 10 π cm$ . The radius is not squared because this is a length.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 $\frac{10}{2} = 5$ 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 ↗

In this lesson

Understand the idea.

Two different measurements

Keep pi until the end

See the method in action.

Radius supplied

Identify the radius and the two formulas.

Calculate the circumference.

Calculate the area.

Diameter supplied

Convert the diameter to a radius.

Substitute the radius into the area formula.

Square the radius and retain pi.

Try it yourself.

Check your understanding

Keep practising