G9, G17, G18Geometry & MeasuresFoundation & HigherGrades 3–7

Circles

Q: How do I remember which formula is area?

Area is in square units, so its formula has a square: r^2. Circumference is a length, so it has r (or d) to the power 1.

Every circle has the same shape, so its circumference and area are fixed multiples of its radius — and the multiple is $π$ , about 3.14159. With $C = π d$ and $A = π r^{2}$ , plus the idea of taking a fraction of a circle, you can handle arcs, sectors, semicircles and running tracks.

45 min AQA GCSE Maths

The big picture

$π$ is simply the number of diameters that fit around a circle's edge. It is irrational, so exact answers are left "in terms of $π$ ", like $25 π$ cm², and decimal answers are always rounded.

Arcs and sectors are fractions of a whole circle. A $9 0^{\circ}$ sector is $\frac{90}{360} = \frac{1}{4}$ of the circle, so its area is a quarter of $π r^{2}$ . That single idea replaces two more formulas.

What you'll learn

Name the parts of a circle
Calculate circumference and area, including in terms of $π$
Find the radius or diameter from a circumference or area
Calculate arc lengths, sector areas and perimeters of sectors (Higher)

Parts of a circle

The $radius$ goes from the centre to the edge; the $diameter$ goes right across through the centre, so $d = 2 r$ . The $circumference$ is the perimeter.

A $chord$ joins two points on the circle; a $tangent$ touches it at one point. An $arc$ is part of the circumference; a $sector$ is a "pizza slice" between two radii; a $segment$ is cut off by a chord.

Radius, sector, chord, segment and tangent.

Circumference and area

Circumference $C = π d = 2 π r$ . Area $A = π r^{2}$ — always use the radius for area.

"In terms of $π$ " means leave $π$ as a symbol: a circle of radius 5 has area $25 π$ cm². Otherwise use the $π$ button and round as instructed.

1(a)

C = π \times 14 = 43.98 \dots

2(b)

r = 7

, so

A = π \times 7^{2} = 49 π

Answer(a)

44.0

cm (b)

49 π

cm²

Your turnSame method, new numbers — try it before moving on.

Find the area of a circle with radius 6.5 cm. Give your answer to 1 decimal place.

Your answer

Your turnSame method, new numbers — try it before moving on.

A circle has circumference 50 cm. Find its radius, to 1 decimal place.

Your answer

Tip — Check whether you are given the radius or the diameter before substituting — using the diameter in $π r^{2}$ makes the area four times too big.

Semicircles and composite shapes

A semicircle has half the area of the full circle. Its perimeter is half the circumference $plus the diameter$ — the straight edge is part of the outline.

1Semicircle radius

= 3

cm. Area

= \frac{1}{2} π \times 9 = 14.14

cm².

2Total area

= 60 + 14.14 = 74.1

cm².

3Perimeter:

10 + 6 + 10

(three straight sides)

+ \frac{1}{2} π \times 6 = 26 + 9.42 = 35.4

cm.

AnswerArea

74.1

cm², perimeter

35.4

Your turnSame method, new numbers — try it before moving on.

Find the perimeter of a semicircle with diameter 12 cm, to 1 decimal place.

Your answer

Arcs and sectors (Higher)

For a sector with angle $θ$ : arc length $= \frac{θ}{360} \times 2 π r$ and sector area $= \frac{θ}{360} \times π r^{2}$ .

The perimeter of a sector is the arc length plus two radii.

1Area

= \frac{135}{360} \times π \times 64 = 24 π

cm².

2Arc

= \frac{135}{360} \times 2 π \times 8 = 6 π

cm.

3Perimeter

= 6 π + 16

cm.

AnswerArea

24 π

cm²; perimeter

6 π + 16

Your turnSame method, new numbers — try it before moving on.

A sector has radius 10 cm and angle $7 2^{\circ}$ . Find its arc length in terms of $π$ .

Your answer

Type π as pi, e.g. 6pi.

One fraction, two formulas

A sector is a fraction of a circle, and the fraction is $\frac{θ}{360}$ . Arc length and sector area both use that same fraction — of the circumference and of the area respectively.

Think like an examiner

Write the unrounded value (e.g. $43.982 \dots$ ) before rounding — it can rescue the method mark.
"In terms of $π$ " answers must contain $π$ ; a decimal loses the mark.
Perimeters of sectors and semicircles need the straight edges added.

Circle formulas

C = π d = 2 π r

Circumference.

A = π r^{2}

Area — use the radius.

\frac{θ}{360} \times 2 π r, \frac{θ}{360} \times π r^{2}

Arc length and sector area.

Watch out for these

Using the diameter in

π r^{2}

Halve the diameter first.

π r^{2}

with

r = 6

(π \times 6)^{2}

Square only the radius:

π \times 36

Perimeter of a semicircle

= \frac{1}{2} π d

only.

Add the diameter.

Arc length using

\frac{θ}{360} \times π r^{2}

Arc is a length: use

2 π r

Stretch yourself

A square of side 10 cm has a quarter circle of radius 10 cm drawn inside it, centred at one corner. Find the area of the square not covered by the quarter circle, in the form $a - bπ$ .

Hint — Subtract the quarter-circle area from the square.

Questions students ask

Should I use 3.14 or the π button?

Use the

π

button unless the question gives a value. Using 3.14 can change the last digit of a rounded answer.

How do I remember which formula is area?

Area is in square units, so its formula has a square:

π r^{2}

. Circumference is a length, so it has

r

(or

d

) to the power 1.

What is the difference between a sector and a segment?

A sector is bounded by two radii and an arc (a pizza slice). A segment is bounded by a chord and an arc.

Key takeaways

$d = 2 r$ ; $C = π d$ ; $A = π r^{2}$ .
In terms of $π$ : keep $π$ as a symbol.
Work backwards by rearranging: $r = \frac{C}{2 π}$ or $r = \frac{A}{π}$ .
Semicircle and sector perimeters include the straight edges.
Arcs and sectors use the fraction $\frac{θ}{360}$ .

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

A circular table top has radius 45 cm. Work out its area. Give your answer to the nearest cm².[2]

Question 2Grade 53 marks

A running track is made from two straight sections of length 100 m and two semicircular ends of diameter 64 m.

Work out the total length of one lap. Give your answer to the nearest metre.[3]

Question 3Grade 74 marks

A sector has an angle of $4 0^{\circ}$ and an arc length of $4 π$ cm. Find the area of the sector, in terms of $π$ .[4]

Unlimited practice

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A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

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Real past-paper questions on Circles, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Circles

Parts of a circle

Circumference and area

Semicircles and composite shapes

Arcs and sectors (Higher)

Practise it the way it is examined

Skill drill

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Related topics

Circles

Parts of a circle

Circumference and area

Semicircles and composite shapes

Arcs and sectors (Higher)

Practise it the way it is examined

Skill drill

Practise this topic

Related topics