POP

C5.3 Circles, Arcs and Sectors

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Every circle is governed by $\pi \approx 3.14159\ldots$ , the ratio between circumference and diameter — no matter the size of the circle, dividing its circumference by its diameter always gives this same constant.

01 — Explanation

What are the circumference and area formulas?

C = \pi d = 2\pi r \qquad A = \pi r^2

Worked example. A circular pond has radius $4.5\text{ m}$ . Find its circumference and area, each to 3 s.f.

C = 2\pi(4.5) = 28.3\text{ m} \ (3\text{ s.f.}) \qquad A = \pi(4.5)^2 = 63.6\text{ m}^2 \ (3\text{ s.f.})

Exam tip

keep the unrounded value on your calculator (or write it with several decimal places) and only round at the very last step. Rounding the radius or an intermediate result too early can shift the final answer outside the accepted tolerance.

02 — Explanation

How do I find the length of an arc?

An arc is a fraction of the whole circumference. At Core tier, the sector angle $\theta$ is always a factor of $360^\circ$ :

\text{arc length} = \frac{\theta}{360^\circ} \times \pi d

Worked example. A sector has radius $9\text{ cm}$ and angle $60^\circ$ . Find the arc length to 3 s.f.

\frac{60}{360} \times \pi \times (2\times 9) = \frac{1}{6}\times 18\pi = 3\pi = 9.42\text{ cm} \ (3\text{ s.f.})

Common mistake

reusing the radius from the area formula without switching to $\pi d$ or $2\pi r$ . Arc length comes from the circumference; always check whether you've been given the radius or the diameter first.

03 — Explanation

How do I find the area of a sector?

A sector is the same fraction of the whole circle's area:

\text{sector area} = \frac{\theta}{360^\circ}\times \pi r^2

Worked example. Using the same sector (radius $9\text{ cm}$ , angle $60^\circ$ ), find the sector area to 3 s.f.

\frac{60}{360}\times \pi \times 9^2 = \frac{1}{6}\times 81\pi = 13.5\pi = 42.4\text{ cm}^2 \ (3\text{ s.f.})

04 — Explanation

How do I find the perimeter of a sector, rather than just its arc length?

A sector's boundary is made of the curved arc plus the two straight radii on either side — a common exam trap is to give only the arc length when the "perimeter" of the whole sector shape was actually requested.

Worked example. A sector has radius $9\text{ cm}$ and angle $60^\circ$ , the same sector as above. Find its perimeter to 3 s.f.

The arc length was already found to be $9.42\text{ cm}$ (3 s.f.). Adding the two straight radii:

\text{perimeter} = 9.42\ldots + 9 + 9 = 27.4\text{ cm} \ (3\text{ s.f.})

Common mistake

stopping after calculating the arc length when the question asks for the sector's full perimeter. Always re-read the question to check whether "arc length" or "perimeter of the sector" was actually requested, and add the two radii whenever it's the latter.

05 — Explanation

What does "leave your answer in terms of π" actually mean?

Some questions ask for an answer written using the symbol $\pi$ itself, rather than a rounded decimal — this gives an exact value with no rounding error at all, which is why exam papers sometimes prefer it for later steps in a longer problem.

Worked example. A sector has radius $12\text{ cm}$ and angle $90^\circ$ . Find its area, leaving your answer in terms of $\pi$ .

\frac{90}{360}\times\pi\times12^2 = \frac{1}{4}\times144\pi = 36\pi\text{ cm}^2

The answer is left as $36\pi\text{ cm}^2$ rather than evaluated to a decimal — multiplying $\pi$ out at this stage would introduce unnecessary rounding, especially if this sector's area is only an intermediate step in a bigger question.

Common mistake

substituting $\pi\approx3.14$ into a calculation when the question explicitly asks for an answer "in terms of $\pi$ ." Doing so turns an exact answer into an inexact one and loses the accuracy mark, even if the decimal value itself is correct.