Revision mat · Arc Length & Sector Area

16 marks

Question mix weighted by 55 real exam questions from 53 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

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) $C=2\pi r=\pi d, \quad A=\pi r^2$

Sector area≈47% of real exam Qs

1The diagram shows a sector of a circle with radius $4$ cm and angle $40^\circ$ at the centre.
Work out the area of the sector.
Give your answer correct to 3 significant figures.

Answer: (3 marks)

Arc length≈16% of real exam Qs

2The diagram shows a sector of a circle with radius $12$ cm and angle $36^\circ$ at the centre.
Work out the length of the arc of the sector.
Give your answer correct to 3 significant figures.

Answer: (3 marks)

Perimeter of a sector≈9% of real exam Qs

3The diagram shows a sector of a circle with radius $6$ cm and angle $30^\circ$ at the centre.
Work out the perimeter of the sector.
Give your answer correct to 3 significant figures.

Answer: (3 marks)

Reverse (find θ or r)≈6% of real exam Qs

4The diagram shows a sector of a circle with radius $6$ cm.
The area of the sector is $6\pi$ cm².
Work out the angle $x$ of the sector.
Give your answer in degrees.

Answer: (3 marks)

★ Exam capstonemixed & other forms ≈22% of real exam Qs

5The diagram shows a sector $OAB$ of a circle, centre $O$, with $OA = OB = 9$ cm and angle $AOB = 120^\circ$. $AB$ is a chord.
Work out the area of the shaded segment (between the chord $AB$ and the arc).
Give your answer correct to 3 significant figures.

Answer: (4 marks)

Mark scheme · Arc Length & Sector Area

Total: 16 marks

Award the marks shown for each correct step, then add up the total out of 16. A method mark counts even if the final answer is wrong.

1The diagram shows a sector of a circle with radius $4$ cm and angle $40^\circ$ at the centre.
Work out the area of the sector.
Give your answer correct to 3 significant figures.[3 marks]

Method

Sector area $= \dfrac{40}{360} \times \pi r^2 = \dfrac{40}{360} \times \pi \times 4^2$.

$= 5.59$ cm² (3 s.f.).

Answer: $5.59$

Marks

✔1 markFraction = $\dfrac{40}{360}$

✔1 markArea = fraction × πr²

✔1 mark5.59 cm²

2The diagram shows a sector of a circle with radius $12$ cm and angle $36^\circ$ at the centre.
Work out the length of the arc of the sector.
Give your answer correct to 3 significant figures.[3 marks]

Method

Arc length $= \dfrac{36}{360} \times 2\pi r = \dfrac{36}{360} \times 2 \times \pi \times 12$.

$= 7.54$ cm (3 s.f.).

Answer: $7.54$

Marks

✔1 markFraction = $\dfrac{36}{360}$

✔1 markArc = fraction × 2πr

✔1 mark7.54 cm

3The diagram shows a sector of a circle with radius $6$ cm and angle $30^\circ$ at the centre.
Work out the perimeter of the sector.
Give your answer correct to 3 significant figures.[3 marks]

Method

Arc length $= \dfrac{30}{360} \times 2\pi \times 6 = 3.14$ cm.

Perimeter $=$ arc $+ 2$ radii $= 3.14 + 2 \times 6 = 15.1$ cm (3 s.f.).

Answer: $15.1$

Marks

✔1 markArc = ($\dfrac{30}{360}$) × 2πr

✔1 markAdd the two radii (+2r)

✔1 mark15.1 cm

4The diagram shows a sector of a circle with radius $6$ cm.
The area of the sector is $6\pi$ cm².
Work out the angle $x$ of the sector.
Give your answer in degrees.[3 marks]

Method

Sector area $= \dfrac{x}{360} \times \pi r^2$, so $6\pi = \dfrac{x}{360} \times \pi \times 6^2$.

Divide by $\pi$: $6 = \dfrac{x}{360} \times 36$.

$x = \dfrac{6 \times 360}{36} = 60^\circ$.

Answer: $60$

Marks

✔1 markArea = (x/360) × πr²

✔1 markDivide by π, then rearrange for x

✔1 mark60°

Method

Sector area $= \dfrac{120}{360} \times \pi \times 9^2 = 84.82$ cm².

Triangle area $= \dfrac{1}{2} r^2 \sin\theta = \dfrac{1}{2} \times 9^2 \times \sin 120^\circ = 35.07$ cm².

Segment $=$ sector $-$ triangle $= 84.82 - 35.07 = 49.7$ cm² (3 s.f.).

Answer: $49.7$

Marks

✔1 markSector area = (θ/360)πr²

✔1 markTriangle area = ½r²sinθ

✔1 markSegment = sector − triangle

✔1 mark49.7 cm²