Circles, Arcs & SectorsCambridge IGCSE Maths: Revision notes

Section 1

How do I find the circumference and area of a circle?

Both formulas are given in the exam, but you must know how to use them:

Circumference: $C = 2\pi r$ (or $C = \pi d$ , since $d = 2r$ )
Area: $A = \pi r^2$

Always check whether you're given the radius or diameter before substituting — a common error is using the diameter as if it were the radius.

Key termscircumferenceradiusdiameter

Common mistake

Using the diameter directly in A = πr² without halving it first gives an area four times too large.

Section 2

What is an arc, and how do I calculate arc length?

An arc is part of a circle's circumference. Arc length is found by treating it as a fraction of the whole circumference, using the angle at the centre, $\theta$ : arc length $= \frac{\theta}{360} \times 2\pi r$ .

Core: the sector angle will always be a factor of 360° (e.g. 90°, 60°, 45°), making the fraction simple. Extended: the angle can be any value, and you may be asked about both the minor arc (shorter) and major arc (longer) of the same circle.

Key termsarcminor arcmajor arc

Example

A circle has radius 6 cm. Find the arc length for a sector angle of 60°. Arc length = 60/360 × 2π(6) = 1/6 × 12π = 2π ≈ 6.28 cm.

Section 3

What is a sector, and how do I calculate sector area?

A sector is the 'pie-slice' region enclosed by two radii and an arc. Its area is found the same way as arc length — as a fraction of the whole circle's area: sector area $= \frac{\theta}{360} \times \pi r^2$ .

At Extended level, this also applies to major sectors (angle greater than 180°), not just minor sectors.

Key termssector

Example

A sector has radius 10 cm and angle 90°. Sector area = 90/360 × π(10)² = 1/4 × 100π = 25π ≈ 78.5 cm².

Section 4

How do I know whether to leave an answer in terms of π?

Exam questions sometimes ask for an answer 'in terms of $\pi$ ', meaning you leave $\pi$ symbolically in your answer rather than using a decimal approximation.

'In terms of π': e.g. $12\pi$ cm
Numerical answer: use the calculator's π button, or 3.142 if no calculator is available, then round sensibly (usually 3 significant figures)

Key termsin terms of π

Exam tip

Read the question wording carefully — writing a decimal when 'in terms of π' is required loses the accuracy mark even if your method is correct.

Section 5

How do I combine circle formulas with perimeter and area of compound shapes?

Many exam questions combine a sector or part-circle with straight-edged shapes (e.g. a semicircle attached to a rectangle).

Method:

Identify which parts are circular (arc/sector) and which are straight
Apply the correct circle formula to the circular part only
Add straight-edge lengths separately for perimeter, or straight-shape areas separately for area
Combine all parts for the total answer

Must Know

Circumference: $C = 2\pi r$ ; Area: $A = \pi r^2$ — both formulas are given
Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr²
Core: sector angle is always a factor of 360°; Extended: any angle, including major sectors/arcs
Never confuse radius and diameter when substituting into formulas
'In terms of π' means leave π symbolic, not converted to a decimal
For compound shapes, treat circular and straight parts separately then combine

That's the notes covered.

Carry on to the next subtopic.