Circular measure: AS Level Mathematics questions with answers
Free Circular measure questions for AS Level Mathematics (9709), part of Pure Mathematics 1. Each question has the answer and the marking points an examiner would look for.
A sector of a circle has radius r cm and angle θ radians. Its perimeter is 18 cm and its area is 20 cm².
(a)
Show that 2r2 − 18r + 40 = 0.
[3]
(b)
Find the possible values of r and the corresponding values of θ.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
shown
2r + rθ = 18 B1 ½r2θ = 20 and θ eliminated M1 given result obtained A1
3
(b)
r = 4, θ = 2.5; r = 5, θ = 1.6
solve the quadratic M1 r = 4 and 5 A1 θ = 2A/r2 for each M1 θ = 2.5 and 1.6 A1
4
Question 2Structured · 7 marksA* challenge
A and B are points on a circle with centre O and radius 12 cm. Angle AOB = π3 radians. The tangent to the circle at A meets the line OB extended at T.
(a)
Find the exact perimeter of the region bounded by AT, TB and the arc AB.
[4]
(b)
Find the exact area of this region.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
12√3 + 12 + 4π cm
AT = r tan θ B1 OT = r ÷ cos θ, so TB = OT − r M1 arc AB = rθ B1 12√3 + 12 + 4π A1
4
(b)
72√3 − 24π cm²
area of triangle OAT = ½ × OA × AT M1 − sector ½r2θ M1 72√3 − 24π A1
3
Question 3Structured · 5 marks
A sector OAB of a circle has radius 6 cm and angle AOB = 1.3 radians. Find (a) the perimeter of the sector, and (b) the area of the segment bounded by the chord AB and the arc AB.
Two circles each have radius 3 cm. The centre of each circle lies on the circumference of the other. The circles meet at P and Q, and their centres are A and B.
(a)
Explain why triangle APB is equilateral and state angle PAQ in radians.
[2]
(b)
Find the exact perimeter of the region common to both circles.
[2]
(c)
Find the exact area of the region common to both circles.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
AP = AB = BP = r; 2π3
all sides equal the radius B1 angle PAQ = 2π3B1
2
(b)
4π cm
two arcs, each 3 × 2π3M1 4π cm A1
2
(c)
9(2π3 − √32) cm²
sector area ½ × 9 × 2π3M1 triangle APQ area ½ × 9 × sin 2π3M1 region = 2(sector − triangle) M1 9(2π3 − √32) cm² A1
4
Question 5Structured · 5 marksA* challenge
Two circles each of radius 5 cm have centres A and B, 5 cm apart.
Find the exact area of the region common to both circles.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
503π − 25√32 cm²
angle at A is 2π3B1 sector area ⅓πr2M1 triangle area ½r2 sin(2π/3) M1 segment = sector − triangle M1 2 segments: (2π/3 − √3/2)r2A1
5
Question 6Structured · 7 marksA* challenge
A chord AB of length 9 cm is drawn in a circle with centre O and radius 6 cm.
(a)
Show that angle AOB is approximately 1.696 radians, correct to 3 decimal places.
[2]
(b)
Find the length of the minor arc AB.
[2]
(c)
Find the area of the major segment.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
1.696
sin(θ/2) = 4.5/6 M1 θ = 1.696 A1
2
(b)
10.2 cm
rθ M1 10.2 cm A1
2
(c)
100 cm²
minor segment ½r2(θ − sin θ) M1 circle area minus minor segment M1 100 cm² A1
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in