Cambridge International AS & A Level Mathematics 9709: Circular measure – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover section 1.4 Circular measure of the Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2026 and 2027. The section is part of Pure Mathematics 1, examined on Paper 1, which is compulsory for AS Level and A Level. A scientific calculator is allowed in every 9709 paper, so none of these is calculator-free. Where a question asks for an exact answer, a decimal will not do. Otherwise give 3 significant figures, and angles in degrees to 1 decimal place.

Learn it first: Circular measure study guide · Recall: Circular measure revision notes · Course hub: Cambridge A Level Mathematics · Checklist: 9709 checklist

In the questions, O is always the centre of the circle and angles without a degree sign are in radians.

Questions

1.

(a) Express 210° in radians, giving your answer as an exact multiple of π. [1]

(b) Convert 0.85 radians to degrees. [1]

2. A sector OAB has radius 7.5 cm and angle AOB = 0.96 radians.

(a) Calculate the length of the arc AB. [1]

(b) Calculate the area of the sector. [2]

3. A sector of a circle has area 54 cm² and arc length 12 cm. Find the radius of the circle and the angle of the sector. [3]

4. A circle has radius 5 cm. Two radii OA and OB form a minor sector with angle AOB = 1.3 radians. Calculate the perimeter of the major sector. [3]

5. A sector OAB has radius 6 cm and angle AOB = 5π/6.

(a) Find the exact area of triangle OAB. [2]

(b) Hence find the exact area of the segment bounded by the chord AB and the arc AB. [2]

6. A circle has radius 12 cm. The minor arc AB has length 15 cm.

(a) Find angle AOB. [1]

(b) Calculate the length of the chord AB. [2]

(c) Calculate the area of the minor segment cut off by AB. [2]

7. A sector has radius r cm and angle θ radians. Its perimeter is equal to the circumference of a circle of radius ½r cm.

(a) Show that θ = π − 2. [2]

(b) Find, in exact form, the ratio of the area of the sector to the area of the circle of radius ½r. [2]

8. A windscreen wiper is modelled as a straight blade that turns about a fixed point O. The part of the blade that clears the glass runs from 15 cm to 50 cm from O. The blade turns through 110°.

(a) Express 110° in radians as an exact multiple of π. [1]

(b) Calculate the length of the path traced by the outer end of the blade. [1]

(c) Calculate the area of glass cleared by the blade. [3]

9. A piece of wire 24 cm long is bent to form the perimeter of a sector with radius r cm and angle θ radians.

(a) Show that the area, A cm², of the sector is given by A = 12r − r². [2]

(b) Express A in the form a − (r − b)², where a and b are constants. Hence state the maximum area of the sector and find the value of θ when the area is a maximum. [3]

10. Two circles each have radius 4 cm. Their centres, A and B, are 4 cm apart. The circles meet at the points P and Q.

(a) Show that angle PAQ = 2π/3. [2]

(b) Find the exact perimeter of the region that lies inside both circles. [2]

(c) Find the exact area of the region that lies inside both circles. [3]

11. A sector OAB has radius 10 cm and angle AOB = 0.9 radians. The point C lies on OA and BC is perpendicular to OA. The region R is bounded by the line CA, the arc AB and the line BC.

(a) Calculate the lengths OC and BC. [2]

(b) Calculate the perimeter of R. [2]

(c) Calculate the area of R. [3]

12. A circle has centre O and radius 8 cm. The point P lies outside the circle. The lines PA and PB are tangents to the circle at A and B, and angle APB = 0.8 radians.

(a) Show that angle AOB = π − 0.8. [2]

(b) Calculate the length PA. [2]

(c) Calculate the area of the region bounded by PA, PB and the minor arc AB. [3]

(d) Calculate the perimeter of this region. [2]

Answers

1. (a) 210 × π/180 = 7π/6 [1]

(b) 0.85 × 180/π = 48.70…, so 48.7° [1]

Examiner insight: an answer such as 210π/180 is not in its simplest form. Simplify the fraction to earn the mark. For (b), 1 d.p. is the standard accuracy for angles in degrees.

2. (a) s = rθ = 7.5 × 0.96 = 7.2 cm [1]

(b) A = ½ × 7.5² × 0.96 [1] = 27 cm² [1]

Examiner insight: the method mark in (b) needs the correct formula with the correct values substituted. Writing ½ × 7.5 × 0.96 (r instead of r²) loses both marks.

3. A = ½r²θ = ½r(rθ) = ½r × 12, so 6r = 54 [1]

r = 9 cm [1]

θ = s/r = 12/9 = 4/3 radians (1.33) [1]

Examiner insight: spotting that rθ is the arc length avoids simultaneous equations. The angle mark can be earned on follow-through from a wrong r if θ = 12/r is used.

4. Reflex angle AOB = 2π − 1.3 = 4.983… [1]

Major arc = 5 × 4.983… = 24.92 cm [1]

Perimeter = 24.92 + 5 + 5 = 34.9 cm [1]

Examiner insight: using 1.3 instead of 2π − 1.3 gets no method marks. The final mark needs both radii added; the arc alone is not the perimeter.

5. (a) Area = ½ × 6 × 6 × sin(5π/6) [1]

= 18 × ½ = 9 cm² [1]

(b) Sector = ½ × 6² × 5π/6 = 15π [1]

Segment = 15π − 9 cm² [1]

Examiner insight: “exact” means sin(5π/6) must be written as ½, not 0.5 read off a calculator with no working. A decimal final answer (38.1) scores the method mark but not the accuracy mark.

6. (a) θ = 15/12 = 1.25 radians [1]

(b) AB = 2 × 12 × sin(0.625) [1] = 14.0 cm [1]

(c) Segment = ½ × 12² × (1.25 − sin 1.25) [1] = 21.7 cm² [1]

Examiner insight: the cosine rule is an equally valid method in (b). In (c), a calculator left in degree mode gives sin 1.25 = 0.0218 and an area of 88.4, which loses the accuracy mark even though the method is right.

7. (a) 2r + rθ = 2π(½r) = πr [1]

Divide by r: 2 + θ = π, so θ = π − 2 [1]

(b) Sector = ½r²(π − 2) and circle = π(½r)² = ¼πr² [1]

Ratio = ½(π − 2) ÷ ¼π = 2(π − 2)/π, that is (2π − 4) : π [1]

Examiner insight: this is a “show that”, so the equation 2r + rθ = πr must appear before the result. Writing θ = π − 2 with no equation scores nothing.

8. (a) 110 × π/180 = 11π/18 [1]

(b) Path length = 50 × 11π/18 = 96.0 cm [1]

(c) Area = ½ × 50² × 11π/18 − ½ × 15² × 11π/18 [1]

= ½ × 2275 × 11π/18 [1]

= 2183.8…, so 2180 cm² (3 s.f.) [1]

Examiner insight: the cleared area is the difference of two sectors. Using ½(50 − 15)² × θ instead of ½(50² − 15²) × θ loses the method mark. Note that 2184 is 4 significant figures; give 2180.

9. (a) Perimeter: 2r + rθ = 24, so rθ = 24 − 2r [1]

A = ½r²θ = ½r(24 − 2r) = 12r − r² [1]

(b) 12r − r² = 36 − (r − 6)² [1]

Maximum A = 36 cm², when r = 6 [1]

θ = (24 − 12)/6 = 2 radians [1]

Examiner insight: the result in (a) is given, so ½r(24 − 2r) must be written out in full. In (b), “hence” means the completed square is the expected route to the maximum.

10. (a) AP = BP = AB = 4, so triangle APB is equilateral and angle PAB = π/3 [1]

By symmetry about AB, angle QAB = π/3, so angle PAQ = 2π/3 [1]

(b) Each arc = 4 × 2π/3 = 8π/3 [1]

Perimeter = 2 × 8π/3 = 16π/3 cm [1]

(c) One segment = ½ × 4² × 2π/3 − ½ × 4² × sin(2π/3) [1]

= 16π/3 − 4√3 [1]

Region = 2 segments = 32π/3 − 8√3 cm² [1]

Examiner insight: in (a) the reason “equilateral” must be stated; a bare “angle PAB = 60°” is not enough for a “show that”. In (c), finding one segment and forgetting to double it loses only the final mark.

11. (a) OC = 10 cos 0.9 = 6.22 cm [1]

BC = 10 sin 0.9 = 7.83 cm [1]

(b) Arc AB = 10 × 0.9 = 9 and CA = 10 − 6.216 = 3.784 [1]

Perimeter = 9 + 7.833 + 3.784 = 20.6 cm [1]

(c) Sector OAB = ½ × 10² × 0.9 = 45 [1]

Triangle OCB = ½ × 6.2161 × 7.8333 = 24.346 [1]

Area of R = 45 − 24.346 = 20.654, so 20.7 cm² [1]

Examiner insight: carry the unrounded OC and BC into (b) and (c). Using 6.22 and 7.83 in (c) gives a triangle of 24.351 and 45 − 24.351 = 20.649, which rounds to 20.6 instead of 20.7 and loses the accuracy mark.

12. (a) Angle OAP = angle OBP = π/2, since a tangent is perpendicular to the radius [1]

Angles in quadrilateral OAPB add to 2π, so angle AOB = 2π − π/2 − π/2 − 0.8 = π − 0.8 [1]

(b) OP bisects angle APB, so angle APO = 0.4 and tan 0.4 = 8/PA [1]

PA = 8/tan 0.4 = 18.9 cm [1]

(c) Kite OAPB = 2 × ½ × 8 × 18.92 = 151.4 [1]

Sector AOB = ½ × 8² × (π − 0.8) = 74.93 [1]

Area = 151.4 − 74.93 = 76.4 cm² [1]

(d) Arc AB = 8 × (π − 0.8) = 18.73 [1]

Perimeter = 2 × 18.92 + 18.73 = 56.6 cm [1]

Examiner insight: in (a) the reason “tangent perpendicular to radius” must be given for the first mark. Parts (c) and (d) allow follow-through from a wrong PA, so a slip in (b) need not cost later method marks.

Where marks are usually lost

• Putting a degree value of θ into s = rθ or ½r²θ. Convert first.
• Working out sin θ in degree mode for a radian angle, as in Question 6(c).
• Missing the two radii from a sector perimeter (Question 4).
• Using the minor angle for a major sector or major arc.
• Rounding an angle or length early and carrying it forward, then losing the final accuracy mark.
• Giving a decimal when the question asks for an exact answer (Questions 5, 7 and 10).
• On a “show that”, not stating the reason for an angle (equilateral triangle, tangent perpendicular to radius) or skipping algebra steps.
• Keeping a value of θ bigger than 2π after solving a quadratic.
• Forgetting to double a segment when a region is made of two equal segments.

Next steps

• Recap formulas and methods in the circular measure revision notes.
• Reread any method you missed in the circular measure study guide.
• Try mixed Paper 1 questions in the Pure Mathematics 1 mixed practice, and quadratic skills in the quadratics practice.
• Take the free 9709 AS diagnostic or use the 9709 self-check question bank.
• Course hub: Cambridge A Level Mathematics · Checklist: 9709 checklist
• Book a free trial class

Official syllabus

Cambridge International, Cambridge International AS & A Level Mathematics 9709 syllabus for 2026 and 2027 (Version 4): Subject content, 1 Pure Mathematics 1 (for Paper 1), section 1.4 Circular measure. Formulae references are to the syllabus’s List of formulae and statistical tables (MF19).