Circles, Arcs & Sectors (Edexcel IGCSE Maths): Revision notes

Section 1

How do I find the circumference of a circle?

The circumference is the perimeter (distance round the outside) of a circle.

$C = \pi d = 2\pi r$

where $d$ is the diameter and $r$ is the radius ( $d = 2r$ ).

A semicircle's perimeter is the curved arc (half the circumference) plus the diameter (the straight edge): $P_{semi} = \pi r + 2r$

A quarter circle's perimeter is a quarter of the circumference plus two radii: $P_{quarter} = \frac{1}{2}\pi r + 2r$

Key termscircumferencediameterradiussemicircle

Common mistake

Do not forget the straight edges when finding the perimeter of a semicircle or quarter circle — only the curved part uses $\pi$ .

Example

Circle with radius 5 cm: $C = 2\pi(5) = 10\pi = 31.4$ cm (3 s.f.).

Section 2

How do I find the area of a circle?

$A = \pi r^2$

Always use the radius in this formula, not the diameter. If you're given the diameter, halve it first.

For a semicircle: $A_{semi} = \frac{1}{2}\pi r^2$

For a quarter circle: $A_{quarter} = \frac{1}{4}\pi r^2$

Key termsarea

Common mistake

A very common error: using $\pi d^2$ instead of $\pi r^2$ . Always check whether you were given the radius or diameter.

Exam tip

Leaving your answer in terms of $\pi$ (e.g. $25\pi$ cm $^2$ ) is exact — only round to decimals if the question asks for it.

Section 3

What is arc length and how do I calculate it?

An arc is a portion of the circumference. Its length depends on the angle $\theta$ (in degrees) it subtends at the centre, as a fraction of the full 360°:

$\text{Arc length} = \frac{\theta}{360} \times 2\pi r$

Think of it as: what fraction of the whole circle's edge does this angle 'cut off'?

Key termsarcsubtend

Think of it like this

Picture a pizza: the arc is the crusty outer edge of a single slice — bigger slice angle means longer crust.

Example

Radius 8 cm, angle 60°: Arc length $= \frac{60}{360} \times 2\pi(8) = \frac{1}{6} \times 16\pi = \frac{8\pi}{3} = 8.38$ cm (3 s.f.).

Section 4

What is sector area and how do I calculate it?

A sector is the pizza-slice region enclosed by two radii and an arc. Like arc length, its area is a fraction of the full circle's area:

$\text{Sector area} = \frac{\theta}{360} \times \pi r^2$

A semicircle is a sector with $\theta = 180°$ ; a quarter circle is a sector with $\theta = 90°$ .

Key termssector

Example

Radius 10 cm, angle 135°: Sector area $= \frac{135}{360} \times \pi(10)^2 = 0.375 \times 100\pi = 37.5\pi = 117.8$ cm $^2$ (1 d.p.).

Exam tip

If a question gives you the arc length or sector area and asks for $\theta$ or $r$ , rearrange the formula — don't try to memorise a separate rearranged version.

Section 5

How do I find the perimeter of a sector?

The perimeter of a sector is not just the arc length — it includes the two straight radii as well:

$P_{sector} = \text{arc length} + 2r = \frac{\theta}{360} \times 2\pi r + 2r$

This is one of the most commonly forgotten steps in exam answers.

Key termsperimeter

Common mistake

Losing marks by giving only the arc length when asked for the perimeter of a sector — always add the two radii.

Example

Radius 6 cm, angle 90°: arc length $= \frac{90}{360} \times 2\pi(6) = 3\pi$ ; perimeter $= 3\pi + 12 = 21.4$ cm (3 s.f.).

Must Know

$C = \pi d = 2\pi r$ ; $A = \pi r^2$ — always use radius for area, diameter for the basic circumference formula
Semicircle perimeter = $\pi r + 2r$ ; semicircle area = $\frac{1}{2}\pi r^2$
Quarter circle perimeter = $\frac{1}{2}\pi r + 2r$ ; quarter circle area = $\frac{1}{4}\pi r^2$
Arc length $= \frac{\theta}{360} \times 2\pi r$ ; sector area $= \frac{\theta}{360} \times \pi r^2$
Sector perimeter = arc length $+ 2r$ (never forget the two radii)
Leave answers in terms of $\pi$ unless told to round, and always check units (length vs area)

That's the notes covered.

Carry on to the next subtopic.