1MA1 · G17–G18Geometry & MeasuresFoundation & HigherGrades 4–7

Circles, Arcs & Sectors

Q: What does "in terms of π" mean?

Leave as a symbol in the answer, e.g. 16 cm², rather than converting to a decimal.

The distance around a circle is always a little over three times its diameter — exactly $\pi$ times. From that one constant come the formulas for circumference and area, and from those come arcs and sectors, which are simply fractions of a whole circle.

45 min Mensuration

The big picture

An arc is part of the circumference, and a sector is part of the area — like a slice of pizza. Both are the same fraction of the whole circle: the angle at the centre divided by $360^{\circ}$ . That means one idea handles every arc and sector question.

Edexcel often asks for answers "in terms of $\pi$ ", which means leaving $\pi$ as a symbol, like $7\pi$ cm. That keeps answers exact and avoids rounding — and is often quicker, because you never need to multiply by 3.14159.

What you'll learn

Calculate the circumference and area of a circle
Calculate arc lengths and sector areas
Find the perimeter of a sector and of compound shapes with circular parts
Give answers in terms of $\pi$ and work backwards to find a radius or angle

Circumference and area

Circumference $C = \pi d = 2\pi r$ . Area $A = \pi r^{2}$ .

Check whether you are given the radius or the diameter; the area formula needs the radius.

In terms of $\pi$ : a circle of radius 5 cm has area $25\pi$ cm² and circumference $10\pi$ cm.

C = 2\pi r

Circumference.

A = \pi r^{2}

Area.

Example 1A circular pond has diameter 12 m. Find its area to 1 decimal place.

1Radius

= 6

A = \pi \times 36 = 113.097\dots

Answer113.1 m²

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

Find the area of a circle with radius 7 cm, to 1 decimal place.

Your answer

To 1 decimal place.

Show worked solution

Solution

1 $A = \pi \times 7^2 = 49\pi$ .
2 $= 153.938\ldots$

Answer: $153.9$ cm²

Tip — Square the radius before multiplying by $\pi$ . $\pi r^{2}$ is $\pi \times r^{2}$ , not $(\pi r)^{2}$ .

Arcs and sectors

A sector with angle $\theta$ at the centre is $\dfrac{\theta}{360}$ of the whole circle.

So arc length $= \dfrac{\theta}{360} \times 2\pi r$ and sector area $= \dfrac{\theta}{360} \times \pi r^{2}$ .

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

l = \dfrac{\theta}{360} \times 2\pi r

Arc length.

A = \dfrac{\theta}{360} \times \pi r^{2}

Sector area.

Example 1A sector has radius 9 cm and angle

140^{\circ}

. Find, in terms of

\pi

, its arc length and area, and find its perimeter to 1 d.p.

1Arc

= \dfrac{140}{360} \times 18\pi = 7\pi

cm.

2Area

= \dfrac{140}{360} \times 81\pi = 31.5\pi

cm².

3Perimeter

= 7\pi + 18 = 39.99\dots

AnswerArc

7\pi

cm, area

31.5\pi

cm², perimeter 40.0 cm

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

A sector has radius 6 cm and angle $100^\circ$ . Find its arc length to 1 decimal place.

Your answer

To 1 decimal place.

Show worked solution

Solution

1Arc $= \frac{100}{360} \times 2\pi \times 6$ .
2 $= 10.47\ldots$ cm.

Answer: $10.5$ cm

Why this works

A semicircle is just a sector with $\theta = 180^{\circ}$ , and a quadrant has $\theta = 90^{\circ}$ . The fraction method covers them without separate formulas.

Compound shapes

Shapes combining rectangles and semicircles, or circles with parts removed, are handled by adding and subtracting areas.

For perimeters, include only the outer edges — not the lines where shapes join.

Example 1A running track is a rectangle 100 m by 60 m with a semicircle on each short end. Find the distance around the track.

1Two straights:

200

2Two semicircles make a full circle of diameter 60:

60\pi

3Total

= 200 + 60\pi = 388.5

Answer

200 + 60\pi \approx 388.5

Working backwards

If the area or arc length is given, substitute into the formula and solve for the radius or the angle.

To find $r$ from an area, divide by $\pi$ then square root.

Example 1A sector has angle

120^{\circ}

and arc length

10\pi

cm. Find its radius.

\dfrac{120}{360} \times 2\pi r = 10\pi

\dfrac{2\pi r}{3} = 10\pi

, so

r = 15

Answer15 cm

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

A sector has angle $60^\circ$ and arc length $4\pi$ cm. Find its radius, in cm.

Your answer

Show worked solution

Solution

1 $\frac{60}{360} \times 2\pi r = 4\pi$ , so $\frac{\pi r}{3} = 4\pi$ .
2 $r = 12$ cm.

Answer: $12$ cm

Tip — When the answer is "in terms of $\pi$ ", divide both sides by $\pi$ early — the $\pi$ cancels and the algebra is simpler.

Think like an examiner

State whether you are using radius or diameter.
Leave answers in terms of $\pi$ when asked; otherwise round as instructed.
Include both radii in a sector perimeter.

Circles

C = 2\pi r = \pi d

Circumference.

A = \pi r^{2}

Area.

\tfrac{\theta}{360} \times 2\pi r

Arc length.

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

Sector area.

Watch out for these

Using the diameter in

\pi r^{2}

Halve the diameter first.

Giving the arc length as the sector perimeter.

Add two radii.

Using

\tfrac{\theta}{180}

as the fraction.

Use

\tfrac{\theta}{360}

Including internal joining lines in a compound perimeter.

Only the outer boundary counts.

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 region inside the square but outside the quarter circle, in terms of $\pi$ and to 1 d.p.

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

Reveal solution

SolutionSquare area

= 100

cm². Quarter circle area

= \tfrac14 \times \pi \times 100 = 25\pi

cm². Region

= 100 - 25\pi \approx 21.5

cm².

\blacksquare

Questions students ask

What does "in terms of π" mean?

Leave

\pi

as a symbol in the answer, e.g.

16\pi

cm², rather than converting to a decimal.

Is a semicircle’s perimeter half the circumference?

No — add the diameter too, the straight edge.

Which value of π should I use?

Use the π button on your calculator unless the question says otherwise.

Key takeaways

Circumference $2\pi r$ ; area $\pi r^{2}$ .
Arcs and sectors are $\tfrac{\theta}{360}$ of the circle.
Sector perimeter $=$ arc $+ 2r$ .
Split compound shapes; count only outer edges for perimeter.
Work backwards by substituting and solving for $r$ or $\theta$ .

Exam-style questions

Practise it the way it is examined

3 original questions · 6 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 41 mark

A circle has diameter 18 cm. Write down its circumference in terms of $\pi$ .[1]
Show mark scheme
- B1 $18\pi$ cm
Answer: $18\pi$ cm

Question 2Grade 63 marks

A running track has two straight sections of 80 m joined by semicircular ends of diameter 50 m. Work out the length of one lap, to 1 decimal place.[3]
Show mark scheme
- M1Two semicircles make a circle: $50\pi$
- M1 $160 + 50\pi$
- A1 $317.1$ m
Answer: $317.1$ m

Question 3Grade 72 marks

A sector of a circle of radius 12 cm has area $24\pi$ cm². Work out the angle of the sector.[2]
Show mark scheme
- M1 $\frac{\theta}{360} \times 144\pi = 24\pi$
- A1 $60^\circ$
Answer: $60^\circ$

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.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 4 lessons