POP

E5.3 Circles, Arcs and Sectors

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01 — Explanation

What are the two core circle formulas?

\text{Circumference} = \pi d = 2\pi r \qquad \qquad \text{Area} = \pi r^2

Both are given in Cambridge's List of Formulas, so you don't need to memorise them — but you do need to know when to use $r$ and when a question gives you $d$ instead.

02 — Explanation

How do I find an arc length or sector area, given the angle at the centre?

An arc or sector is simply a fraction of the whole circle, and that fraction is the sector's centre-angle out of the full $360°$ :

\text{Arc length} = \frac{\theta}{360°} \times \pi d \qquad \qquad \text{Sector area} = \frac{\theta}{360°} \times \pi r^2

03 — Explanation

How do I tell a minor sector/arc from a major sector/arc?

The minor arc/sector corresponds to the smaller centre angle $\theta$ ; the major arc/sector corresponds to the reflex angle $360° - \theta$ . If you're asked for the major sector, either use $360°-\theta$ in the formula directly, or find the minor sector first and subtract it from the full circle's area.

04 — Explanation

What does "leave your answer in terms of \pi" actually require?

It means don't evaluate $\pi$ as a decimal — keep it as a symbol throughout, only combining it with the numerical coefficient at the very end (e.g. $18\pi\text{ cm}^2$ , not $56.5\text{ cm}^2$ ). This gives an exact answer with no rounding error.

05 — Explanation

How do I find the angle at the centre if I'm given the arc length or sector area instead?

Rearrange the relevant formula to make $\theta$ the subject. From the sector area formula: $\theta = \dfrac{\text{sector area}}{\pi r^2} \times 360°$ . From the arc length formula: $\theta = \dfrac{\text{arc length}}{\pi d} \times 360°$ . This "reverse" version is just as likely to appear as the forward version, so practise substituting into both directions rather than only ever solving for arc length or sector area.

06 — Explanation

How do I find the perimeter of a sector, not just its area?

The perimeter of a sector is not the arc length alone — it includes the two straight radii as well, since a sector is bounded by an arc and two radii meeting at the centre:

\text{Perimeter of sector} = \text{arc length} + 2r

Forgetting to add the two radii is an extremely common way to lose a mark on an otherwise correct sector calculation.

Exam tip

Always check whether the question has given you the radius or the diameter — substituting a diameter directly into the $\pi r^2$ area formula (instead of halving it first) is one of the most frequent errors at this level.

07 — Worked example

Worked example

A sector has a radius of $9\text{ cm}$ and a centre angle of $80°$ . Find (a) the arc length and (b) the sector area, in terms of $\pi$ .

(a) Arc length

\text{Arc length} = \frac{80}{360} \times \pi (18) = \frac{2}{9} \times 18\pi = 4\pi \text{ cm}

(b) Sector area

\text{Sector area} = \frac{80}{360} \times \pi (9)^2 = \frac{2}{9} \times 81\pi = 18\pi \text{ cm}^2

08 — Worked example

Worked example: finding the angle and the perimeter

A sector has a radius of $6\text{ cm}$ and an arc length of $5\pi\text{ cm}$ . Find (a) the angle at the centre, and (b) the perimeter of the sector.

(a) Rearrange the arc length formula: $\theta = \dfrac{\text{arc length}}{\pi d} \times 360° = \dfrac{5\pi}{\pi(12)} \times 360° = \dfrac{5}{12} \times 360° = 150°$ .
(b) Perimeter $= \text{arc length} + 2r = 5\pi + 2(6) = 5\pi + 12 \text{ cm}$ .