Unlike many other formulas, $A=\pi r^2$ and $C=2\pi r$ are not given on the formula sheet — and every sector/arc formula is built directly from them. Without these memorised, the derived formulas become an easily-confused jumble of symbols.

An arc is simply a fraction of the full circumference, where that fraction is the central angle $\theta$ out of the full $360°$:

Example: $r=8\text{ cm}$, $\theta=135°$: Arc Length $=\dfrac{135}{360}\times2\pi(8) = 6\pi \approx18.8\text{ cm}$ (3 s.f.).

The same fraction-of-the-whole logic applies to area:

Example: $r=7.1\text{ cm}$, $\theta=110°$: Area $=\dfrac{110}{360}\times\pi\times7.1^2\approx48.4\text{ cm}^2$ (1 d.p.).

A sector's boundary has three parts: two straight radii and the curved arc. Forgetting the two radii — and reporting only the arc length — is one of the most common ways marks are lost here.

Example: $r=5.2\text{ cm}$, $\theta=67°$. Arc length $\approx6.08\text{ cm}$ (keep unrounded). Perimeter $=(2\times5.2)+6.08\approx16.5\text{ cm}$ (3 s.f.).

Rearrange the sector or arc formula as you would any equation. Example: area $=50\text{ cm}^2$, $r=8\text{ cm}$, find $\theta$:

Worked Example

A sector has radius $12\text{ cm}$ and central angle $45°$. Find the arc length, leaving your answer in terms of $\pi$.

1. Arc Length $=\dfrac{45}{360}\times2\pi(12)$.
2. $\dfrac{45}{360}=\dfrac18$, so Arc Length $=\dfrac18\times24\pi=3\pi\text{ cm}$.

Some questions combine a sector with a triangle or another circle — for example, a segment (the region between a chord and its arc) is found by taking the sector's area and subtracting the triangle formed by the two radii and the chord. Always identify which simple shapes have been combined, calculate each one's area separately using the formula that fits it, then add or subtract as the diagram requires. Treat the angle $\theta$ and radius $r$ as the anchor values you extract first — every other calculation in the question builds from them.