Arcs and sectors Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Arcs and sectors questions. See exactly how to solve problems on arc length, fraction of a circle, exact answers in terms of pi, sector area.

arc lengthfraction of a circleexact answers in terms of pisector areasimplifyingperimeter of a sector
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
The diagram shows a sector of a circle of radius 99 cm with an angle of 60∘60^\circ at the centre. Work out the arc length of the sector. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the arc length of a sector.

    arc length=θ360×2πr\text{arc length} = \frac{\theta}{360} \times 2\pi r

    An arc is part of the circumference. A sector with an angle of θ\theta at the centre takes up θ360\frac{\theta}{360} of the full turn, so its arc takes up the same fraction of the circumference 2πr2\pi r.

  2. Work out what fraction of the whole circle the sector is.

    60360=16\frac{60}{360} = \frac{1}{6}

    A full turn is 360∘360^\circ, so an angle of 60∘60^\circ at the centre gives 16\frac{1}{6} of the whole circle.

  3. Take that fraction of the circumference and state the arc length.

    arc length=16×2×π×9=3π cm\text{arc length} = \frac{1}{6} \times 2 \times \pi \times 9 = 3\pi\text{ cm}

    The arc is 16\frac{1}{6} of the circumference 2πr2\pi r, so it is 3π3\pi cm — exact, in terms of π\pi.

Answer
arc length=3π cm\text{arc length} = 3\pi\text{ cm}
Question 2
2 markseasy
The diagram shows a sector of a circle of radius 88 cm with an angle of 45∘45^\circ at the centre. Work out the arc length of the sector. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the arc length of a sector.

    arc length=θ360×2πr\text{arc length} = \frac{\theta}{360} \times 2\pi r

    An arc is part of the circumference. A sector with an angle of θ\theta at the centre takes up θ360\frac{\theta}{360} of the full turn, so its arc takes up the same fraction of the circumference 2πr2\pi r.

  2. Work out what fraction of the whole circle the sector is.

    45360=18\frac{45}{360} = \frac{1}{8}

    A full turn is 360∘360^\circ, so an angle of 45∘45^\circ at the centre gives 18\frac{1}{8} of the whole circle.

  3. Take that fraction of the circumference and state the arc length.

    arc length=18×2×π×8=2π cm\text{arc length} = \frac{1}{8} \times 2 \times \pi \times 8 = 2\pi\text{ cm}

    The arc is 18\frac{1}{8} of the circumference 2πr2\pi r, so it is 2π2\pi cm — exact, in terms of π\pi.

Answer
arc length=2π cm\text{arc length} = 2\pi\text{ cm}
Question 3
2 markseasy
The diagram shows a sector of a circle of radius 66 cm with an angle of 120∘120^\circ at the centre. Work out the arc length of the sector. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the arc length of a sector.

    arc length=θ360×2πr\text{arc length} = \frac{\theta}{360} \times 2\pi r

    An arc is part of the circumference. A sector with an angle of θ\theta at the centre takes up θ360\frac{\theta}{360} of the full turn, so its arc takes up the same fraction of the circumference 2πr2\pi r.

  2. Work out what fraction of the whole circle the sector is.

    120360=13\frac{120}{360} = \frac{1}{3}

    A full turn is 360∘360^\circ, so an angle of 120∘120^\circ at the centre gives 13\frac{1}{3} of the whole circle.

  3. Take that fraction of the circumference and state the arc length.

    arc length=13×2×π×6=4π cm\text{arc length} = \frac{1}{3} \times 2 \times \pi \times 6 = 4\pi\text{ cm}

    The arc is 13\frac{1}{3} of the circumference 2πr2\pi r, so it is 4π4\pi cm — exact, in terms of π\pi.

Answer
arc length=4π cm\text{arc length} = 4\pi\text{ cm}
Question 4
1 markeasy
The diagram shows a sector of a circle of radius 1010 cm with an angle of 90∘90^\circ at the centre. Work out the arc length of the sector. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the arc length of a sector.

    arc length=θ360×2πr\text{arc length} = \frac{\theta}{360} \times 2\pi r

    An arc is part of the circumference. A sector with an angle of θ\theta at the centre takes up θ360\frac{\theta}{360} of the full turn, so its arc takes up the same fraction of the circumference 2πr2\pi r.

  2. Work out what fraction of the whole circle the sector is.

    90360=14\frac{90}{360} = \frac{1}{4}

    A full turn is 360∘360^\circ, so an angle of 90∘90^\circ at the centre gives 14\frac{1}{4} of the whole circle.

  3. Take that fraction of the circumference and state the arc length.

    arc length=14×2×π×10=5π cm\text{arc length} = \frac{1}{4} \times 2 \times \pi \times 10 = 5\pi\text{ cm}

    The arc is 14\frac{1}{4} of the circumference 2πr2\pi r, so it is 5π5\pi cm — exact, in terms of π\pi.

Answer
arc length=5π cm\text{arc length} = 5\pi\text{ cm}
Question 5
2 markseasy
The diagram shows a sector of a circle of radius 66 cm with an angle of 90∘90^\circ at the centre. Work out the area of the sector. Give your answer in terms of π\pi.

Worked solution

  1. Write down the formula for the area of a sector.

    area=θ360×πr2\text{area} = \frac{\theta}{360} \times \pi r^2

    A sector is part of the whole circle. It takes up θ360\frac{\theta}{360} of the full turn, so it takes up the same fraction of the area πr2\pi r^2.

  2. Work out what fraction of the whole circle the sector is.

    90360=14\frac{90}{360} = \frac{1}{4}

    A full turn is 360∘360^\circ, so an angle of 90∘90^\circ at the centre gives 14\frac{1}{4} of the whole circle.

  3. Take that fraction of the area of the circle and state the answer.

    area=14×π×62=9π cm2\text{area} = \frac{1}{4} \times \pi \times 6^2 = 9\pi\text{ cm}^2

    The sector is 14\frac{1}{4} of the circle of area πr2\pi r^2, so its area is 9π9\pi square cm — exact, in terms of π\pi.

Answer
area=9π cm2\text{area} = 9\pi\text{ cm}^2

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