Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Arcs and sectors. unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
An arc is part of the circumference, and a sector is the 'slice' between two radii. You work out arc lengths, sector areas and perimeters as a fraction of the whole circle, sometimes in terms of π, and work backwards to find an angle or a radius.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Name the parts of a circleRadius, diameter, circumference, chord, tangent, arc, sector and segment.
5
Calculate the length of an arcArc length = \(\frac{\theta}{360} \times 2\pi r\), a fraction of the circumference.
5
Calculate the area of a sectorSector area = \(\frac{\theta}{360} \times \pi r^2\), a fraction of the area of the circle.
5
Find the perimeter of a sectorAdd the two radii to the arc length.
5
Give answers in terms of πLeave π as a symbol and simplify, e.g. \(\frac{40}{360} \times \pi \times 9^2 = 9\pi\).
Notes
Parts of a circle
An arc is part of the circumference. A sector is the region between two radii and an arc.
A chord is a straight line joining two points on the circle. A segment is the region between a chord and an arc.
A tangent is a straight line that touches the circle at one point.
The minor arc or sector is the smaller one (angle less than 180°); the major one is the larger.
Arc length and sector area
A sector with angle θ at the centre is \(\frac{\theta}{360}\) of the whole circle.
Arc length = \(\frac{\theta}{360} \times 2\pi r\) (that fraction of the circumference).
Sector area = \(\frac{\theta}{360} \times \pi r^2\) (that fraction of the area).
Example: r = 6 cm and θ = 120°. Arc length = \(\frac{120}{360} \times 12\pi = 4\pi\) = 12.6 cm. Sector area = \(\frac{120}{360} \times 36\pi = 12\pi\) = 37.7 cm².
Perimeter of a sector = arc length + 2 radii. In the example: \(4\pi + 12\) = 24.6 cm.
For a major sector, use 360° − θ as the angle.
Working backwards
If you are given the arc length or area, substitute everything you know and solve.
Cheatsheet
Circumference = 2πr = πd; area of a circle = πr²
Arc length = \(\frac{\theta}{360} \times 2\pi r\)
Sector area = \(\frac{\theta}{360} \times \pi r^2\)
Example. OAB is a sector of a circle with centre O. OA = OB = 9 cm and angle AOB = 140°. Work out the perimeter of the sector. Give your answer correct to 1 decimal place.
Show the model answer
Arc AB = \(\frac{140}{360}\) × 2 × π × 9 = 21.99… cm (M1) Perimeter = 21.99… + 9 + 9 (M1) = 40.0 cm (A1)
For answers in terms of π, cancel the numbers and write π at the end, e.g. 12π.
Where marks are lost
Forgetting to add the two radii to find the perimeter of a sector.
Using the diameter where the formula needs the radius.
Using θ ÷ 180 or θ ÷ 100 instead of θ ÷ 360.
Using the minor angle when the question asks about the major sector.
Using 3.14 instead of the π button, which can change the last digit of your answer.
Exam technique
'Give your answer in terms of π' means leave π in: an answer like 12π is exact.
Write the full calculation (e.g. 64/360 × π × 7.5²) before the answer to earn the method mark.
Give units: cm for lengths and perimeters, cm² for areas.
For a compound shape, list which edges make up the perimeter before adding them.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
The diagram shows a sector of a circle, centre \(O\). The radius of the circle is 15.6 cm and the angle of the sector is 130°. Diagram NOT accurately drawn
[object Object]
Work out the area of the sector. Give your answer correct to 3 significant figures.[2]
Show the answer and mark scheme
Answer: \(276\) cm²
M1 for \(\frac{130}{360} \times \pi \times 15.6^2\)
A1 for 276 (276.08...)
Worked solution: Area \(= \frac{130}{360} \times \pi \times 15.6^2\) \(= 276.08\ldots\) = 276 cm² (3 s.f.)
Question 2Medium3 marks
The diagram shows a sector of a circle, centre \(O\). The radius of the circle is 15 cm and the angle of the sector is 72°. Diagram NOT accurately drawn
[object Object]
Work out the perimeter of the sector. Give your answer in terms of \(\pi\).[3]
Show the answer and mark scheme
Answer: \(30 + 6\pi \) cm
M1 for the arc length \(\frac{72}{360} \times 2 \times \pi \times 15\) (= \(6\pi \))
M1 for adding 30 (two radii) to their arc length
A1 for \(30 + 6\pi \) oe
Worked solution: Arc length \(= \frac{72}{360} \times 2\pi \times 15 = 6\pi \) cm Perimeter \(= 15 + 15 + 6\pi = 30 + 6\pi \) cm