Arc Length & Sector Area · Knowledge Organiser

Higher · grade 5–7 · non-calculator or calculator

Key formulas

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

A sector: the region between two radii and an arc. $\theta$ is the angle at the centre.

Worked examples

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.)

Key words

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.

Common mistakes

✗ 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).

Key facts

• 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.

Remember

• 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.

Retrieval starter · Arc Length & Sector Area

Fill the gaps from memory, then check

Cover the organiser. Fill in as much as you can from memory, then turn it over (or use the answers strip) to check and correct in a different colour.

A · Write each formula

Arc length

Sector area

Sector perimeter

Whole circle (reference)

B · Define each key word

Arc

Sector

Segment

Subtend

Chord

C · Complete the facts

A full circle is °, so a sector is the fraction of the whole circle.

A semicircle has $\theta=$ ° and a quarter circle has $\theta=$ °.

A sector's perimeter is the arc length $+$ .

Unless it says "in terms of $\pi$", round answers to significant figures.

Answers: Arc length: $L=\dfrac{\theta}{360}\times 2\pi r$ · Sector area: $A=\dfrac{\theta}{360}\times \pi r^2$ · Sector perimeter: $P=\dfrac{\theta}{360}\times 2\pi r \; + \; 2r$ · Whole circle (reference): $C=2\pi r=\pi d, \quad A=\pi r^2$ · Words: 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 · Facts: 360°, $\tfrac{\theta}{360}$, 180°, 90°, two radii ($2r$), 3 s.f.