Mathematics 9709 · syllabus 1.4
Cambridge A Level Mathematics 9709 — Circular measure topical past papers
Radians, arc length and sector area, and how to combine them with triangles to find the perimeters and areas of shaded regions.
- Questions
- 104
- Years
- 2009–2026
- With mark scheme
- 98%
- Questions 2022–2026
- 33
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What Circular measure covers
A radian is the angle at the centre of a circle made by an arc equal in length to the radius, so a full turn is 2π radians and π radians is 180°. Measuring angles in radians makes the sector formulae simple: arc length s = rθ and area A = ½r²θ, both with θ in radians.
Questions combine sectors with triangles and other shapes. You may need the area of a segment (a sector minus a triangle), the perimeter of a shaded region made of arcs and straight edges, or a missing angle found with right-angled trigonometry or the cosine rule. Exact answers in terms of π and surds are often asked for, so exact trigonometric values are worth knowing well.
Circular measure is section 1.4 of Paper 1. Radians are used from here on: in the trigonometric graphs and equations of 1.5, and in all the calculus of trigonometric functions in Papers 2 and 3, where the derivative of sin x is cos x only when x is measured in radians.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 1.4 Circular measure
What you should be able to do
- Convert between degrees and radians
- Find arc lengths and sector areas with the angle in radians
- Find the area of a segment as a sector minus a triangle
- Find the perimeter of a region made of arcs and straight lines
- Use right-angled trigonometry, the sine rule or the cosine rule to find missing lengths and angles
Key ideas
- π radians = 180°, so 1 radian ≈ 57.3°
- Arc length s = rθ, with θ in radians
- Sector area A = ½r²θ, with θ in radians
- Triangle area = ½ab sin C, so a segment with angle θ at the centre has area ½r²(θ − sin θ)
- Exact values in radians: sin(π/6) = ½, cos(π/6) = √3/2, sin(π/4) = cos(π/4) = 1/√2, tan(π/3) = √3
How Circular measure has been examined
From 2022 to 2026, Circular measure was on Paper 1 · Pure Mathematics 1 in 11 of 11 exam series, worth about 9% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Pure Mathematics 1 | 33 | 9% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 7.
Filed in Momentum under: Radians.
A worked example
A circle has centre O and radius 6 cm. Points A and B lie on the circle with angle AOB = π/3 radians. Find, in exact form, the perimeter and the area of the segment bounded by the chord AB and the minor arc AB.
- Arc AB = rθ = 6 × π/3 = 2π cm.
- Triangle OAB has OA = OB = 6 cm with an angle of π/3 between them, so it is equilateral and AB = 6 cm.
- Perimeter of the segment = arc + chord = (2π + 6) cm.
- Sector area = ½r²θ = ½ × 36 × π/3 = 6π cm².
- Triangle area = ½ × 6 × 6 × sin(π/3) = 18 × √3/2 = 9√3 cm², so the segment's area is 6π − 9√3 cm².
Answer: Perimeter (2π + 6) cm; area (6π − 9√3) cm², about 3.26 cm²
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Using s = rθ with θ in degrees: the formula only holds in radians, so 60° must become π/3 first.
- Leaving the calculator in degree mode while finding sin θ for an angle given in radians, which gives a completely different value.
- Adding only the arcs for a perimeter and leaving out the straight edges, radii or chords, that close the shape.
- Using the sector area for a segment, or subtracting the wrong triangle: the segment is the sector minus the triangle formed by the two radii and the chord.
Exam technique
- Mark every known length and angle on the diagram, and shade the region asked for, before writing any formula.
- Give exact answers in terms of π and surds when the question says 'exact'; otherwise give 3 significant figures, keeping full calculator values until the end.
- When an angle has to be found first, find it in radians and carry the unrounded value into the arc or sector formula.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.