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 $\pi$ , about 3.14159. With $C = \pi d$ and $A = \pi r^2$ , 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

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

Arcs and sectors are fractions of a whole circle. A $90^\circ$ sector is $\frac{90}{360} = \frac{1}{4}$ of the circle, so its area is a quarter of $\pi r^2$ . 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 $\pi$
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 $\textbf{radius}$ goes from the centre to the edge; the $\textbf{diameter}$ goes right across through the centre, so $d = 2r$ . The $\textbf{circumference}$ is the perimeter.

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

Radius, sector, chord, segment and tangent.

Circumference and area

Circumference $C = \pi d = 2\pi r$ . Area $A = \pi r^2$ — always use the radius for area.

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

Example 1A circle has diameter 14 cm. Find (a) its circumference to 1 d.p. (b) its area in terms of

\pi

1(a)

C = \pi \times 14 = 43.98\ldots

2(b)

r = 7

, so

A = \pi \times 7^2 = 49\pi

Answer

(a)

44.0

cm (b)

49\pi

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

Show worked solution

Solution

1 $A = \pi \times 6.5^2 = \pi \times 42.25$ .
2 $= 132.73\ldots$ , so $132.7$ cm².

Answer: $132.7$ cm²

Example 2A circle has area

81\pi

cm². Find its circumference in terms of

\pi

\pi r^2 = 81\pi

, so

r^2 = 81

and

r = 9

C = 2\pi \times 9

Answer

18\pi

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

Show hint

$2\pi r = 50$ .

Show worked solution

Solution

1 $r = \frac{50}{2\pi} = 7.957\ldots$
2To 1 d.p.: $8.0$ cm.

Answer: $8.0$ cm

Tip — Check whether you are given the radius or the diameter before substituting — using the diameter in $\pi r^2$ 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 $\textbf{plus the diameter}$ — the straight edge is part of the outline.

Example 1A shape is a 10 cm by 6 cm rectangle with a semicircle on one of the 6 cm sides. Find its area and perimeter to 1 d.p.

1Semicircle radius

= 3

cm. Area

= \frac{1}{2} \pi \times 9 = 14.14

cm².

2Total area

= 60 + 14.14 = 74.1

cm².

3Perimeter:

10 + 6 + 10

(three straight sides)

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

cm.

Answer

Area

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

Show worked solution

Solution

1Curved part: $\frac{1}{2} \times \pi \times 12 = 18.85$ cm.
2Add the diameter: $18.85 + 12 = 30.85$ , so $30.8$ cm (30.849…).

Answer: $30.8$ cm

Arcs and sectors (Higher)

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

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

Example 1A sector has radius 8 cm and angle

135^\circ

. Find its area and perimeter, in terms of

\pi

1Area

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

cm².

2Arc

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

cm.

3Perimeter

= 6\pi + 16

cm.

Answer

Area

24\pi

cm²; perimeter

6\pi + 16

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

A sector has radius 10 cm and angle $72^\circ$ . Find its arc length in terms of $\pi$ .

Your answer

Type π as pi, e.g. 6pi.

Show worked solution

Solution

1 $\frac{72}{360} = \frac{1}{5}$ .
2Arc $= \frac{1}{5} \times 2\pi \times 10 = 4\pi$ cm.

Answer: $4\pi$ cm

Example 2A sector of radius 9 cm has area

27\pi

cm². Find its angle.

\frac{\theta}{360} \times 81\pi = 27\pi

\frac{\theta}{360} = \frac{1}{3}

, so

\theta = 120^\circ

Answer

120^\circ

One fraction, two formulas

A sector is a fraction of a circle, and the fraction is $\frac{\theta}{360}$ . 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. $43.982\ldots$ ) before rounding — it can rescue the method mark.
"In terms of $\pi$ " answers must contain $\pi$ ; a decimal loses the mark.
Perimeters of sectors and semicircles need the straight edges added.

Circle formulas

C = \pi d = 2\pi r

Circumference.

A = \pi r^2

Area — use the radius.

\tfrac{\theta}{360} \times 2\pi r, \quad \tfrac{\theta}{360} \times \pi r^2

Arc length and sector area.

Watch out for these

Using the diameter in

\pi r^2

Halve the diameter first.

\pi r^2

with

r = 6

(\pi \times 6)^2

Square only the radius:

\pi \times 36

Perimeter of a semicircle

= \frac{1}{2}\pi d

only.

Add the diameter.

Arc length using

\frac{\theta}{360} \times \pi r^2

Arc is a length: use

2\pi 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 $a - b\pi$ .

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

Reveal solution

SolutionSquare:

100

cm². Quarter circle:

\frac{1}{4} \times \pi \times 100 = 25\pi

cm². Uncovered area

= 100 - 25\pi

cm² (about 21.5 cm²).

\blacksquare

Questions students ask

Should I use 3.14 or the π button?

Use the

\pi

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:

\pi 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

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

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]
Show mark scheme
- M1 $\pi \times 45^2$
- A1 $6362$ cm² (6361.7…)
Answer: $6362$ cm²

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]
Show mark scheme
- M1The two semicircles form a full circle: $\pi \times 64 = 201.06\ldots$
- M1Adds the straights: $200 + 201.06$
- A1 $401$ m
Answer: $401$ m

Question 3Grade 74 marks

A sector has an angle of $40^\circ$ and an arc length of $4\pi$ cm. Find the area of the sector, in terms of $\pi$ .[4]
Show mark scheme
- M1 $\frac{40}{360} \times 2\pi r = 4\pi$
- A1 $r = 18$
- M1 $\frac{40}{360} \times \pi \times 18^2$
- A1 $36\pi$ cm²
Answer: $36\pi$ cm²

Unlimited practice

Skill drill

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

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A circle has radius 4 cm. Its diameter is…
- 8 cm
- 2 cm
- 16 cm
- 12.6 cm
Show answer and explanation
Answer: 8 cm
$d = 2r$ .
Area of a circle with radius 3 cm, in terms of $\pi$ ?
- $9\pi$ cm²
- $6\pi$ cm²
- $3\pi$ cm²
- $36\pi$ cm²
Show answer and explanation
Answer: $9\pi$ cm²
$\pi \times 3^2$ .
Circumference of a circle with diameter 10 cm (1 d.p.)?
- 31.4 cm
- 78.5 cm
- 62.8 cm
- 15.7 cm
Show answer and explanation
Answer: 31.4 cm
$\pi \times 10$ .
Which line touches a circle at exactly one point?
- tangent
- chord
- radius
- diameter
Show answer and explanation
Answer: tangent
A tangent just touches the circle.
A circle has area $64\pi$ . Its radius is…
- $8$
- $32$
- $64$
- $4$
Show answer and explanation
Answer: $8$
$r^2 = 64$ .
Area of a semicircle with radius 10 cm, in terms of $\pi$ ?
- $50\pi$ cm²
- $100\pi$ cm²
- $10\pi$ cm²
- $25\pi$ cm²
Show answer and explanation
Answer: $50\pi$ cm²
Half of $100\pi$ .
Arc length of a $90^\circ$ sector with radius 6 cm?
- $3\pi$ cm
- $9\pi$ cm
- $6\pi$ cm
- $12\pi$ cm
Show answer and explanation
Answer: $3\pi$ cm
$\frac{1}{4} \times 12\pi$ .
Area of a $60^\circ$ sector with radius 12 cm?
- $24\pi$ cm²
- $4\pi$ cm²
- $144\pi$ cm²
- $12\pi$ cm²
Show answer and explanation
Answer: $24\pi$ cm²
$\frac{1}{6} \times 144\pi$ .

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