Arc length $L=\dfrac{\theta}{360}\times 2\pi r$the curved edge only Sector area $A=\dfrac{\theta}{360}\times \pi r^2$the "pizza slice" region Sector perimeter $P=\dfrac{\theta}{360}\times 2\pi r \; + \; 2r$arc $+$ two radii Whole circle (reference) $C=2\pi r=\pi d, \quad A=\pi r^2$a sector is a fraction of these
Arc length · Sector: $r=5$ cm, $\theta=72°$.
$L=\dfrac{72}{360}\times 2\pi\times 5$
$=\dfrac{1}{5}\times 10\pi = 2\pi$
$=6.28$ cm (3 s.f.)
Sector area · Same sector: $r=5$ cm, $\theta=72°$.
$A=\dfrac{72}{360}\times \pi\times 5^2$
$=\dfrac{1}{5}\times 25\pi = 5\pi$
$=15.7$ cm² (3 s.f.)
Arc: part of the circumference: the curved edge of a sector. Sector: region bounded by two radii and an arc (a "slice"). Segment: region between a chord and an arc. Subtend: the angle $\theta$ at the centre "opens out" the arc. Chord: a straight line joining two points on the circle.
✗ Using the diameter in $2\pi r$
✓ the formula uses the radius $r$ (halve the diameter first).
✗ Forgetting the $\dfrac{\theta}{360}$ fraction
✓ that gives the whole circle. Always scale by $\dfrac{\theta}{360}$.
✗ Swapping the formulas
✓ arc uses $2\pi r$; area uses $\pi r^2$. Check the units ($\text{cm}$ vs $\text{cm}^2$).
✗ Perimeter $=$ arc only
✓ a sector's perimeter is the arc $+\,2r$ (the two straight radii).
• A full circle is $360°$; a sector is the fraction $\dfrac{\theta}{360}$ of the whole circle.
• Semicircle: $\theta=180°$ (a half). Quarter circle: $\theta=90°$.
• Give lengths/areas to 3 s.f., unless the question says "in terms of $\pi$", then leave $\pi$ in and simplify the fraction.
• Write the fraction $\dfrac{\theta}{360}$ down first, every single time.
• "Perimeter" $=$ arc $+$ 2 radii; "arc length" $=$ just the curved bit.
• In terms of $\pi$: simplify the fraction and leave $\pi$ (e.g. $2\pi$); don't multiply it out.
mathedup.co.uk · sheet H7UI