G17 - Circumference and area of circles | AQA GCSE Mathematics

A circle needs only one length to fix its size. In G17 you will use radius and diameter to work out circumference and area, keep exact answers in terms of $\pi$ , and combine circles with straight edges in perimeter and area problems.

Circumference from radius or diameter

A running track has two straight sections, each $30\,\mathrm{m}$ long, joined by two semicircular ends of diameter $20\,\mathrm{m}$ . Is one lap $60+20\pi$ metres or $60+40\pi$ metres? We need to count how much of a circle the ends actually make.

You already know

A radius joins the centre to the circle; a diameter crosses the circle through its centre. The diameter is twice the radius. An answer in terms of $\pi$ keeps $\pi$ rather than replacing it with a decimal.

The circumference is the distance all the way around a circle. If you divide any circle's circumference by its diameter, you get the same number, $\pi$ . A small coin and a large circular pond have the same ratio. That is why one formula works at every size.

[DIAGRAM: asset_name: Radius diameter circumference and area; asset_slug: aqa8300_gd_circle_measures; caption: Circumference measures the curved boundary and area measures the surface inside; description: A pale blue circle has centre O, a clay diameter d across it and a sage radius r from its centre to its circumference. Text gives C equals pi d equals 2 pi r, and A equals pi r squared. Not drawn accurately.]
Circumference measures the curved boundary and area measures the surface inside

Circle circumference

For radius $r$ and diameter $d=2r$ ,

C=2\pi r=\pi d

Use the same length unit for $C$ , $r$ and $d$ .

On the 2027 formulae sheet.

Decide whether the supplied length crosses the whole circle or only reaches its centre before choosing a substitution.

Worked example 1: a radius is supplied

Question. A circle has radius $4.5\,\mathrm{cm}$ . Work out its circumference. Give your answer to $3$ significant figures. Calculator allowed.

Plan. Use $C=2\pi r$ , keeping $\pi$ in the calculator until the final rounding.

\begin{aligned} C&=2\times\pi\times4.5 &&\text{substitute the radius}\\ &=9\pi &&\text{exact circumference}\\ &=28.27433388\ldots &&\text{calculator value} \end{aligned}

Why use the factor $2$ ? The given length is a radius, half the diameter.

Answer. $28.3\,\mathrm{cm}$ to $3$ significant figures.

Check. $\pi$ is a little more than $3$ , so the circumference should be a little more than $3\times9=27\,\mathrm{cm}$ .

Change only the way the same circle's size is given: supply its diameter instead. The circumference must stay unchanged.

Worked example 2: a diameter is supplied

Question. A circle has diameter $9\,\mathrm{cm}$ . Work out its circumference. Give your answer in terms of $\pi$ . Non-calculator.

Plan. Multiply the diameter directly by $\pi$ .

\begin{aligned} C&=\pi\times9 &&\text{use the diameter formula}\\ &=9\pi &&\text{collect the numerical factor} \end{aligned}

Why not multiply by $2$ again? The full diameter has already been supplied.

Answer. $9\pi\,\mathrm{cm}$ .

Check. Half the diameter is $4.5\,\mathrm{cm}$ , the radius in Worked example 1.

Both formulae give the same boundary length when their inputs describe the same circle.

Watch out

Wrong: a diameter of $9\,\mathrm{cm}$ gives $C=2\pi\times9$ . That treats the diameter as a radius and doubles the answer. Right: $C=\pi\times9$ .

Read the next supplied length carefully before substituting.

Circle areas and unknown radius

A circle's area depends on the square of its radius, so finding area needs a different calculation from finding circumference.

Imagine cutting a circle into many thin sectors and alternating them into a long strip. As the pieces get thinner, the strip approaches a rectangle with height $r$ and length half the circumference, $\pi r$ . Multiplying these gives $\pi r^2$ .

Circle area

For radius $r$ ,

A=\pi r^2

The area uses square units. If a diameter is given, halve it before squaring.

On the 2027 formulae sheet.

The order matters: square the radius, then multiply by $\pi$ . Squaring the whole product $\pi r$ would square $\pi$ too.

Worked example 3: area from radius

Question. A circle has radius $6\,\mathrm{cm}$ . Work out its area. Give your answer to $3$ significant figures. Calculator allowed.

Plan. Square $6$ , multiply by $\pi$ , then round once.

\begin{aligned} A&=\pi\times6^2 &&\text{circle area formula}\\ &=36\pi &&\text{square the radius first}\\ &=113.0973355\ldots &&\text{calculator value} \end{aligned}

Why use $6^2$ ? Area measures two dimensions, so the radius contributes twice as a factor.

Answer. $113\,\mathrm{cm}^2$ to $3$ significant figures.

Check. The circle fits inside a square of side $12\,\mathrm{cm}$ , whose area is $144\,\mathrm{cm}^2$ . The circle's area is smaller.

Change the given measurement from radius to circumference. Recover the radius before calculating area.

Worked example 4: area from circumference

Question. A circle has circumference $12\pi\,\mathrm{cm}$ . Work out its area. Give your answer in terms of $\pi$ . Non-calculator.

Plan. Solve the circumference formula for $r$ , then use the area formula.

\begin{aligned} 2\pi r&=12\pi &&\text{use the circumference}\\ r&=6 &&\text{divide by }2\pi\\ A&=\pi\times6^2=36\pi &&\text{use the recovered radius} \end{aligned}

Why not put $12\pi$ into the area formula? It is a circumference, not a radius.

Answer. $36\pi\,\mathrm{cm}^2$ .

Check. The radius is the same as in Worked example 3, so both areas agree.

If area is supplied instead, reverse the squaring with a square root. The radius is positive because it represents a length.

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G17 - Circumference and area of circles

Circumference from radius or diameter

Circle areas and unknown radius

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