Papy

Arc Length and Sector Area - Edexcel GCSE Higher Maths

Typically 3-5 marks per paper, more where a segment or a compound shape is set · Spec 1MA1

A sector is the slice of a circle held between two radii and the arc that joins their ends. Spec reference G18 on Edexcel's 1MA1 asks you to calculate arc lengths, areas of sectors and the angles that produce them, and G18 is one of the Higher-only references, so a Foundation candidate works with whole circles and stops there. One idea runs through all of it: a sector whose angle at the centre is θ\theta takes up θ360\frac{\theta}{360} of its circle, so the arc is that fraction of the circumference and the sector area is that fraction of the circle's area.

Neither formula is printed on the Higher Tier Exam Aid supplied with the 2025, 2026 and 2027 papers. The sheet gives the circumference and the area of a circle, and that is the design: Edexcel expects you to take θ360\frac{\theta}{360} of something you have been handed rather than to recall θ360×2πr\frac{\theta}{360} \times 2\pi r as a separate fact. On Paper 1 the angle divides into 360 cleanly and the instruction reads 'give your answer in terms of π\pi', which wants 6π6\pi written down and not 18.85. Papers 2 and 3 allow a calculator, so the radius is untidy, the answer is asked for to 3 significant figures, and a question can afford to run backwards from a given area to the angle.

The perimeter of a sector is where the marks are actually decided. Examiner reports return to the same loss year after year: the arc is worked out correctly and then handed in as the perimeter, so the accuracy mark goes because nobody added the two straight edges. Build the habit of three lines, fraction then arc then arc plus 2r2r, and check the units, because a perimeter in cm2^2 is not a perimeter. Beyond that, practise the reverse direction, where an arc or an area is given and the angle or radius has to be extracted, and the segment, which is the sector with the triangle between the two radii removed and needs 12absin⁡C\frac{1}{2}ab\sin C from the formulae sheet.

Worked example

OAB is a sector of a circle with centre O. OA and OB are radii of length 15 cm. Angle AOB = 72∘72^\circ. Work out the perimeter of the sector OAB. Give your answer correct to 1 decimal place.
[4 marks]
  1. 72360=15\frac{72}{360} = \frac{1}{5}, so the sector is one fifth of the whole circle.
    M1: the fraction of the circle, written down before anything is substituted
  2. The circumference of the whole circle is 2×π×15=30π2 \times \pi \times 15 = 30\pi cm, so arc AB=15×30π=6π=18.8495…AB = \frac{1}{5} \times 30\pi = 6\pi = 18.8495\ldots cm.
    M1: a correct method for the arc; this formula is not on the Exam Aid, it is built from the circumference that is
  3. The perimeter runs along the arc and back down both straight edges: 18.8495…+15+15=48.8495…18.8495\ldots + 15 + 15 = 48.8495\ldots cm.
    M1: both radii added to their arc, which is the step the question is set to test
  4. Perimeter =48.8= 48.8 cm, correct to 1 decimal place.
    A1: 48.8, rounded once at the end; an answer of 18.8 is the arc on its own and earns nothing here

Practice questions

These are original questions written in Edexcel 1MA1 Higher style. There are no diagrams, so read each description carefully: check whether you have been given a radius or a diameter, and whether the question wants the arc alone or the whole perimeter. Where a question says the answer is kπk\pi, type the value of kk on its own, and where a rounding is stated, keep the full value on your calculator until the final line.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [2 marks]
    A sector of a circle has radius 9 cm and an angle of 40∘40^\circ at the centre. The length of the arc of the sector is kπk\pi cm. Write down the value of kk.
  2. Question 2 Warm-up [2 marks]
    A sector of a circle has radius 6 cm and an angle of 90∘90^\circ at the centre. Work out the area of the sector. Give your answer in cm2^2, correct to 1 decimal place.
  3. Question 3 Exam pace [3 marks]
    A sector of a circle of radius 9 cm has an area of 40 cm2^2. Work out the angle at the centre of the sector. Give your answer in degrees, correct to 1 decimal place.
  4. Question 4 Exam pace [3 marks]
    A sector of a circle has radius 6 cm and a reflex angle of 210∘210^\circ at the centre. Work out the area of the sector. Give your answer in cm2^2, correct to 3 significant figures.
  5. Question 5 Exam pace [2 marks]
    A sector of a circle has radius 6 cm and an angle of 150∘150^\circ at the centre. Which of these is the area of the sector?
    Answer options for question 5
  6. Question 6 Stretch [5 marks]
    A and B are points on a circle with centre O and radius 14 cm. Angle AOB = 100∘100^\circ. The minor segment is the region bounded by the chord AB and the minor arc AB. Work out the perimeter of the minor segment. Give your answer in cm, correct to 1 decimal place.

Common mistakes examiners see

  • Giving the arc length when the question asked for the perimeter of the sector. A sector of radius 9 cm with an angle of 40∘40^\circ is answered as 2π2\pi, or 6.28 cm, when the perimeter is 2π+18=24.32\pi + 18 = 24.3 cm.

    Trace the boundary with your finger: curved edge, straight edge, straight edge. Write 'arc + 2r' at the top of the working, and if the sector is not a semicircle the perimeter must be more than double the radius.

  • Swapping the two formulae, so the arc is found with πr2\pi r^2 or the area with 2πr2\pi r. For radius 6 cm and angle 150∘150^\circ that turns an area of 15π15\pi cm2^2 into 5π5\pi, which is the arc length.

    Check the power of rr against the units you have been asked for. An area is measured in cm2^2, so the formula must contain r2r^2; a length is measured in cm, so the formula contains a single rr.

  • Putting the diameter into the formula where the radius belongs. A pizza described as 30 cm across gives a slice area of 18×π×302=353\frac{1}{8} \times \pi \times 30^2 = 353 cm2^2 rather than 88.4 cm2^2, four times too big.

    Write r=…r = \ldots on its own line before you substitute, and halve anything the question calls a diameter or describes as a width across. Questions about plates, cakes, ponds and fans nearly always quote the diameter.

  • Rearranging the reverse questions the wrong way round. Given an arc of 10 cm on a circle of radius 12 cm, writing θ=2π×1210×360\theta = \frac{2\pi \times 12}{10} \times 360 instead of θ=102π×12×360\theta = \frac{10}{2\pi \times 12} \times 360, and producing an angle far larger than 360.

    Set it up as an equation in θ\theta first, θ360×75.398…=10\frac{\theta}{360} \times 75.398\ldots = 10, then divide. An angle bigger than 360∘360^\circ or a radius bigger than the arc it carries is a sign the division went upside down.

  • Handling the segment wrongly: subtracting the sector from the triangle, or working out the triangle as 12×\frac{1}{2} \times base ×\times height using the two radii, which are not a base and a perpendicular height.

    Segment = sector minus triangle, in that order, because the triangle sits inside the sector and the answer must be positive. The triangle has two known sides and the angle between them, so use 12absin⁡C\frac{1}{2}ab\sin C with aa and bb both equal to the radius, and keep the unrounded sector value on the display while you subtract.

Frequently asked questions

Is arc length and sector area on the Foundation paper?
No. G18 is a Higher-only reference on Edexcel 1MA1, so sectors are examined on Higher Papers 1, 2 and 3 and nowhere at Foundation. Foundation candidates still need the circumference and area of a full circle, and they meet the vocabulary of arc, chord and sector, but they are not asked to take a fraction of a circle.
Are the arc length and sector area formulas given in the Edexcel maths exam?
No. The Higher Tier Exam Aid issued with the papers for the 2025, 2026 and 2027 series gives the circumference of a circle and the area of a circle, and nothing about sectors. You are expected to multiply those by θ360\frac{\theta}{360} yourself, so arc length =θ360×2πr= \frac{\theta}{360} \times 2\pi r and sector area =θ360×πr2= \frac{\theta}{360} \times \pi r^2 are both recall.
How do you find the perimeter of a sector?
Work out the arc length, then add the two radii: perimeter =θ360×2πr+2r= \frac{\theta}{360} \times 2\pi r + 2r. The two straight edges are worth a mark of their own on the mark scheme and are the most common omission on this question. A perimeter is a length, so the answer is in cm or m, never cm2^2.
How many marks are sector questions worth on Edexcel Higher?
A single arc or area calculation is usually 2-3 marks in the first half of a paper. A perimeter, a segment, a compound shape or a question that works backwards from a given area to the angle typically runs to 4-5 marks later on. Across a paper of 80 marks expect roughly 3-5 marks from this topic.

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