Question Simulator

IGCSE Additional Mathematics (0606)

Circular measure Banker

9 questions · about 5% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 12 · Q6 (a, b)8 marksArc length and sector area

2025

  • 2025 Oct/Nov · Paper 22 · Q2 (a, b(i), b(ii))8 marksArc length and sector area
  • 2025 May/June · Paper 12 · Q10 (a, b)8 marksArc length and sector area

2024

  • 2024 Oct/Nov · Paper 12 · Q3 (a, b, c)7 marksArc length and sector area
  • 2024 May/June · Paper 12 · Q6 (a, b, c)9 marksArc length and sector area

2023

  • 2023 Oct/Nov · Paper 12 · Q10 (a, b)7 marksArc length and sector area
  • 2023 May/June · Paper 12 · Q4 (a, b)7 marksArc length and sector area

2022

  • 2022 Oct/Nov · Paper 12 · Q7 (a, b, c)7 marksArc length and sector area
  • 2022 May/June · Paper 22 · Q9 (a, b)7 marksArc length and sector area

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Arc length and sector area · Core · Paper 2 style · calculator · 4 marks
  1. (a) Express 135° in radians, giving your answer as an exact multiple of π. [1]
  2. (b) Express 2.4 radians in degrees, correct to 1 decimal place. [1]
  3. (c) A sector has radius 5 cm and angle 2.4 radians. Find its area. [2]
Show answer
  1. (a) 3π/4
    B1
  2. (b) 137.5°
    2.4 × 180/π B1
  3. (c) 30 cm²
    ½ × 25 × 2.4 M1 = 30 A1
Q2Arc length and sector area · Core · Paper 1 style · no calculator · 4 marks
  1. (a) A sector of a circle has radius 8 cm and angle 0.75 radians. Find the arc length, the perimeter and the area of the sector. [4]
Show answer
  1. (a) arc 6 cm, perimeter 22 cm, area 24 cm²
    s = rθ = 6 B1; perimeter 6 + 16 = 22 B1; area ½ × 64 × 0.75 M1 = 24 A1
Q3Arc length and sector area · Core · Paper 1 style · no calculator · 4 marks
  1. (a) A sector of radius 6 cm has area 30 cm². Find the angle of the sector in radians. [2]
  2. (b) Find the perimeter of the sector. [2]
Show answer
  1. (a) 5/3 radians
    ½ × 36 × θ = 30 M1; θ = 5/3 A1
  2. (b) 22 cm
    arc = 6 × 5/3 = 10 M1; 10 + 12 = 22 A1
Q4Arc length and sector area · Core · Paper 1 style · no calculator · 4 marks
  1. (a) A sector of radius 8 cm has perimeter 28 cm. Find the angle of the sector and its area. [4]
Show answer
  1. (a) θ = 1.5 radians, area 48 cm²
    arc = 28 − 16 = 12 M1; θ = 12/8 = 1.5 A1; area ½ × 64 × 1.5 M1 = 48 A1
Q5Arc length and sector area · Core · Paper 2 style · calculator · 3 marks
  1. (a) A chord AB of a circle with centre O and radius 10 cm subtends an angle of 1.2 radians at O. Find the area of the minor segment cut off by AB, correct to 3 significant figures. [3]
Show answer
  1. (a) 13.4 cm²
    sector 60 B1; triangle ½ × 100 × sin 1.2 = 46.6 M1; 60 − 46.6 = 13.4 A1
Q6Arc length and sector area · Core · Paper 2 style · calculator · 4 marks

A round pizza from a restaurant in Karachi has diameter 30 cm and is cut into 8 equal sectors.

  1. (a) Find the area of the top of one slice, correct to 1 decimal place. [2]
  2. (b) Find the perimeter of one slice, correct to 1 decimal place. [2]
Show answer
  1. (a) 88.4 cm²
    angle π/4 B1; ½ × 15² × π/4 = 88.4 B1
  2. (b) 41.8 cm
    arc 15 × π/4 = 11.78 M1; 30 + 11.78 = 41.8 A1
Q7Arc length and sector area · Core · Paper 1 style · no calculator · 4 marks

ABCD is a square of side 10 cm. An arc of a circle with centre A and radius 10 cm joins B to D.

  1. (a) Find the exact area of the region inside the square but outside the quarter-circle. [2]
  2. (b) Find the exact perimeter of this region. [2]
Show answer
  1. (a) 100 − 25π cm²
    quarter circle ¼π(100) = 25π M1; 100 − 25π A1
  2. (b) 20 + 5π cm
    arc BD = 10 × π/2 = 5π M1; BC + CD = 20, total 20 + 5π A1
Q8Arc length and sector area · Core · Paper 1 style · no calculator · 3 marks
  1. (a) A sector has arc length 15 cm and area 75 cm². Find the radius and the angle of the sector. [3]
Show answer
  1. (a) r = 10 cm, θ = 1.5 radians
    ½r(rθ) = ½ × r × 15 = 75 M1; r = 10 A1; θ = 1.5 A1
Q9Arc length and sector area · Core · Paper 1 style · no calculator · 4 marks
  1. (a) The minute hand of a clock is 9 cm long. Find the exact distance moved by the tip of the hand in 20 minutes. [2]
  2. (b) Find the exact area swept by the hand in this time. [2]
Show answer
  1. (a) 6π cm
    angle 2π/3 B1; 9 × 2π/3 = 6π B1
  2. (b) 27π cm²
    ½ × 81 × 2π/3 M1 = 27π A1
Q10Arc length and sector area · Core · Paper 2 style · calculator · 5 marks
  1. (a) A door 0.9 m wide swings open through an angle of 110°. Find the area of floor swept by the door, correct to 3 significant figures. [3]
  2. (b) Find the distance moved by the outer edge of the door, correct to 3 significant figures. [2]
Show answer
  1. (a) 0.778 m²
    110° = 1.920 radians B1; ½ × 0.81 × 1.920 M1 = 0.778 A1
  2. (b) 1.73 m
    0.9 × 1.920 M1 = 1.73 A1
Q11Arc length and sector area · Standard · Paper 1 style · no calculator · 7 marks

A circle has centre O and radius 12 cm. The points A and B lie on the circle and angle AOB = 2π/3 radians. Do not use a calculator in this question.

  1. (a) Find the exact length of the chord AB. [2]
  2. (b) Find the exact perimeter of the minor segment cut off by AB. [2]
  3. (c) Find the exact area of the minor segment. [3]
Show answer
  1. (a) 12√3 cm
    2 × 12 sin(π/3) M1 = 12√3 A1
  2. (b) 8π + 12√3 cm
    arc = 12 × 2π/3 = 8π B1; + 12√3 B1
  3. (c) 48π − 36√3 cm²
    sector ½ × 144 × 2π/3 = 48π B1; triangle ½ × 144 × sin(2π/3) = 36√3 B1; 48π − 36√3 B1
Q12Arc length and sector area · Standard · Paper 2 style · calculator · 6 marks

The bob of a clock pendulum is on the end of a string of length 1.2 m. In one swing the string turns through an angle of 0.3 radians.

  1. (a) Find the length of the arc through which the bob moves in one swing. [2]
  2. (b) Find the straight-line distance between the two extreme positions of the bob, correct to 3 significant figures. [2]
  3. (c) Find the area swept out by the string in one swing. [2]
Show answer
  1. (a) 0.36 m
    1.2 × 0.3 M1 = 0.36 A1
  2. (b) 0.359 m
    2 × 1.2 sin 0.15 M1 = 0.359 A1
  3. (c) 0.216 m²
    ½ × 1.44 × 0.3 M1 = 0.216 A1
Q13Arc length and sector area · Standard · Paper 2 style · calculator · 5 marks

A car windscreen wiper arm OB is 50 cm long and pivots at O. The rubber blade covers the part of the arm from 15 cm to 50 cm from O. The arm turns through an angle of 2.1 radians.

  1. (a) Find the area of windscreen wiped by the blade, correct to the nearest cm². [3]
  2. (b) Find the perimeter of the wiped region. [2]
Show answer
  1. (a) 2389 cm²
    ½ × 2.1 × (50² − 15²) M1 M1 = 2388.75 A1
  2. (b) 206.5 cm
    arcs 2.1 × 50 = 105 and 2.1 × 15 = 31.5; straight edges 2 × 35 M1; 105 + 31.5 + 70 = 206.5 A1
Q14Arc length and sector area · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) A sector of a circle of radius r cm and angle θ radians has perimeter 20 cm and area 24 cm². Show that r² − 10r + 24 = 0. [3]
  2. (b) Hence find the two possible pairs of values of r and θ. [3]
Show answer
  1. (a) shown
    rθ + 2r = 20 B1; ½r²θ = 24 B1; ½r(20 − 2r) = 24 ⇒ r² − 10r + 24 = 0 B1
  2. (b) r = 4, θ = 3 or r = 6, θ = 4/3
    r = 4 or 6 B1; θ = 12/4 = 3 B1; θ = 8/6 = 4/3 B1
Q15Arc length and sector area · Standard · Paper 2 style · calculator · 7 marks

OAB is a sector of a circle with centre O, radius 9 cm and angle AOB = 1.4 radians. The point C lies on OA with OC = 5 cm.

  1. (a) Find the length of CB, correct to 3 significant figures. [2]
  2. (b) Find the perimeter of the region bounded by AC, CB and the arc AB. [2]
  3. (c) Find the area of this region. [3]
Show answer
  1. (a) 9.52 cm
    CB² = 81 + 25 − 90 cos 1.4 M1; 9.52 A1
  2. (b) 26.1 cm
    arc 12.6, AC 4 M1; 12.6 + 4 + 9.52 = 26.1 A1
  3. (c) 34.5 cm²
    sector ½ × 81 × 1.4 = 56.7 B1; triangle OCB ½ × 5 × 9 sin 1.4 = 22.2 M1; 34.5 A1
Q16Arc length and sector area · Standard · Paper 2 style · calculator · 5 marks

A circle has centre O and radius 7 cm. AB is a chord of length 10 cm.

  1. (a) Find angle AOB in radians, correct to 4 significant figures. [2]
  2. (b) Find the area of the minor segment cut off by AB, correct to 3 significant figures. [3]
Show answer
  1. (a) 1.591 radians
    sin(½AOB) = 5/7 M1; AOB = 1.591 A1
  2. (b) 14.5 cm²
    sector ½ × 49 × 1.591 = 38.98 B1; triangle ½ × 49 sin 1.591 = 24.5 M1; 14.5 A1
Q17Arc length and sector area · Standard · Paper 2 style · calculator · 7 marks

AB is a diameter of a semicircle with centre O and radius 10 cm. C is a point on the arc such that angle AOC = 2 radians.

  1. (a) Find the length of the chord AC, correct to 3 significant figures. [2]
  2. (b) Find the area of the segment cut off by AC, correct to 3 significant figures. [3]
  3. (c) Find the area of triangle OBC. [2]
Show answer
  1. (a) 16.8 cm
    2 × 10 sin 1 M1 = 16.8 A1
  2. (b) 54.5 cm²
    sector ½ × 100 × 2 = 100 B1; triangle ½ × 100 sin 2 = 45.5 M1; 54.5 A1
  3. (c) 45.5 cm²
    angle BOC = π − 2 M1; ½ × 100 × sin(π − 2) = 45.5 A1
Q18Arc length and sector area · Standard · Paper 2 style · calculator · 6 marks

A park in Islamabad has a flower bed in the shape of a sector of a circle of radius 4.5 m and angle 100°.

  1. (a) Convert 100° to radians, correct to 4 significant figures. [1]
  2. (b) Find the area of the flower bed, correct to 3 significant figures. [2]
  3. (c) Edging costing Rs 350 per metre is fixed along the whole boundary. Find the total cost, to the nearest rupee. [3]
Show answer
  1. (a) 1.745 radians
    B1
  2. (b) 17.7 m²
    ½ × 4.5² × 1.745 M1 = 17.7 A1
  3. (c) Rs 5899
    arc 4.5 × 1.745 = 7.854 M1; perimeter 16.854 A1; × 350 = 5899 A1
Q19Arc length and sector area · Standard · Paper 1 style · no calculator · 4 marks

The region R lies between two arcs with the same centre O and angle θ radians: the inner arc has radius 5 cm and the outer arc has radius 8 cm, and the ends are joined by straight lines along the radii. The area of R is 19.5 cm².

  1. (a) Find the value of θ. [2]
  2. (b) Find the perimeter of the region. [2]
Show answer
  1. (a) θ = 1
    ½θ(8² − 5²) = 19.5 M1; θ = 1 A1
  2. (b) 19 cm
    arcs 8 + 5, straight edges 3 + 3 M1; 19 A1
Q20Arc length and sector area · Standard · Paper 1 style · no calculator · 4 marks

Three identical cylindrical pipes, each of radius 4 cm, are placed so that each touches the other two. A tight band is wrapped once around the three pipes.

  1. (a) Explain why each arc of the band in contact with a pipe subtends an angle of 2π/3 radians at the centre of that pipe. [2]
  2. (b) Find the exact length of the band. [2]
Show answer
  1. (a) the straight parts are tangents perpendicular to radii; angle = 2π − π/2 − π/2 − π/3 = 2π/3
    radii ⟂ tangents M1; 2π/3 A1
  2. (b) 24 + 8π cm
    three straight parts each 8 cm M1; three arcs total 2π × 4 = 8π A1
Q21Arc length and sector area · Standard · Paper 2 style · calculator · 4 marks

Two places lie on the same line of longitude, at latitudes 24.9° N and 34.0° N. Model the Earth as a sphere of radius 6370 km.

  1. (a) Find the angle, in radians, between the two radii from the centre of the Earth to the two points, correct to 4 significant figures. [2]
  2. (b) Find the distance between the two points measured along the surface of the Earth, correct to 3 significant figures. [2]
Show answer
  1. (a) 0.1588 radians
    9.1° B1; × π/180 = 0.1588 B1
  2. (b) 1010 km
    6370 × 0.1588 M1 = 1010 A1
Q22Arc length and sector area · Standard · Paper 1 style · no calculator · 5 marks

OAB is a sector of a circle with centre O, radius 6 cm and angle AOB = 5π/6 radians. Do not use a calculator in this question.

  1. (a) Find the exact area of the sector OAB. [2]
  2. (b) Find the exact area of the minor segment cut off by the chord AB. [3]
Show answer
  1. (a) 15π cm²
    ½ × 36 × 5π/6 M1 = 15π A1
  2. (b) 15π − 9 cm²
    triangle ½ × 36 × sin(5π/6) = 9 M1 A1; 15π − 9 A1
Q23Arc length and sector area · Challenge · Paper 2 style · calculator · 7 marks

A circle has centre O and radius 5 cm. The point T is 13 cm from O. The lines TA and TB are tangents to the circle at A and B.

  1. (a) Show that angle AOB = 2.352 radians, correct to 3 decimal places. [2]
  2. (b) Find the area of the region bounded by TA, TB and the minor arc AB. [3]
  3. (c) Find the perimeter of this region. [2]
Show answer
  1. (a) shown
    cos AOT = 5/13 M1; AOB = 2 cos⁻¹(5/13) = 2.352 A1
  2. (b) 30.6 cm²
    kite OATB area = 2 × ½ × 5 × 12 = 60 B1; sector ½ × 25 × 2.352 = 29.4 M1; 30.6 A1
  3. (c) 35.8 cm
    AT = 12 B1; 24 + 5 × 2.352 = 35.8 B1
Q24Arc length and sector area · Challenge · Paper 1 style · no calculator · 6 marks

Two circles, each of radius 6 cm, have centres P and Q with PQ = 6 cm. The circles intersect at A and B. Do not use a calculator in this question.

  1. (a) Show that the angle subtended at each centre by the common chord is 2π/3 radians. [2]
  2. (b) Find the exact area of the region common to both circles. [4]
Show answer
  1. (a) shown
    triangle formed by the two centres and an intersection point is equilateral M1; angle at centre 2 × π/3 = 2π/3 A1
  2. (b) 24π − 18√3 cm²
    one segment = ½ × 36 × (2π/3 − sin(2π/3)) M1; = 12π − 9√3 A1; double it M1; 24π − 18√3 A1
Q25Arc length and sector area · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) A piece of wire of length 40 cm is bent to form the boundary of a sector of radius r cm. Show that the area A cm² of the sector is given by A = 20r − r². [2]
  2. (b) Find the value of r for which A is a maximum, and find the angle of the sector and the maximum area. [4]
Show answer
  1. (a) shown
    arc = 40 − 2r B1; A = ½r(40 − 2r) = 20r − r² B1
  2. (b) r = 10, θ = 2 radians, A = 100 cm²
    dA/dr = 20 − 2r = 0 M1; r = 10 A1; θ = 20/10 = 2 A1; A = 100 A1
Q26Arc length and sector area · Challenge · Paper 1 style · no calculator · 6 marks

A school in Lahore designs a badge from an equilateral triangle ABC of side 6 cm. Each side is replaced by a circular arc whose centre is the opposite vertex (so arc BC has centre A, and so on). Do not use a calculator in this question.

  1. (a) Find the exact perimeter of the badge. [2]
  2. (b) Find the exact area of the badge. [4]
Show answer
  1. (a) 6π cm
    each arc 6 × π/3 = 2π M1; 3 × 2π = 6π A1
  2. (b) 18π − 18√3 cm²
    sector ½ × 36 × π/3 = 6π B1; triangle (√3/4) × 36 = 9√3 B1; area = 3 sectors − 2 triangles M1; 18π − 18√3 A1
Q27Arc length and sector area · Challenge · Paper 1 style · no calculator · 8 marks

OAB is a sector of a circle with centre O, radius 12 cm and angle AOB = π/3 radians. The tangent to the arc at A meets OB extended at T. Do not use a calculator in this question.

  1. (a) Find the exact length of AT and of BT. [3]
  2. (b) Find the exact perimeter of the region bounded by AT, TB and the arc AB. [2]
  3. (c) Find the exact area of this region. [3]
Show answer
  1. (a) AT = 12√3 cm, BT = 12 cm
    OA ⟂ AT, AT = 12 tan(π/3) = 12√3 M1 A1; OT = 12/cos(π/3) = 24, BT = 12 A1
  2. (b) 12√3 + 12 + 4π cm
    arc 12 × π/3 = 4π B1; total B1
  3. (c) 72√3 − 24π cm²
    triangle OAT ½ × 12 × 12√3 = 72√3 B1; sector 24π B1; 72√3 − 24π B1
Q28Arc length and sector area · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) A circle of radius ρ cm is drawn inside a sector of radius 12 cm and angle π/3 radians so that it touches both straight edges and the arc. Show that ρ = 4. [3]
  2. (b) Find the exact area of the part of the sector outside the circle. [2]
Show answer
  1. (a) shown
    centre on bisector, distance from O = 12 − ρ M1; ρ = (12 − ρ) sin(π/6) M1; ρ = 4 A1
  2. (b) 8π cm²
    sector 24π, circle 16π M1; 8π A1
Q29Arc length and sector area · Challenge · Paper 2 style · calculator · 6 marks

A sector OAB has centre O, radius r cm and angle AOB = 0.8 radians. The perimeter of the minor segment bounded by the chord AB and the arc AB is 12 cm.

  1. (a) Show that r = 7.60, correct to 3 significant figures. [3]
  2. (b) Find the area of the minor segment, correct to 3 significant figures. [3]
Show answer
  1. (a) shown
    chord = 2r sin 0.4 B1; r(0.8 + 2 sin 0.4) = 12 M1; r = 7.60 A1
  2. (b) 2.39 cm²
    ½r²(0.8 − sin 0.8) M1 M1 = 2.39 A1
Q30Arc length and sector area · Challenge · Paper 1 style · no calculator · 7 marks

OABC is a rhombus with sides 8 cm and angle AOC = 2π/3 radians. An arc of a circle with centre O and radius 8 cm joins A to C. Do not use a calculator in this question.

  1. (a) Show that B lies on the arc AC. [2]
  2. (b) Find the exact area of the region inside the sector but outside the rhombus. [3]
  3. (c) Find the exact perimeter of this region. [2]
Show answer
  1. (a) triangle OAB is equilateral, so OB = 8
    angle OAB = π − 2π/3 = π/3 with OA = AB = 8 M1; so OB = 8 A1
  2. (b) 64π/3 − 32√3 cm²
    sector ½ × 64 × 2π/3 = 64π/3 B1; rhombus 64 sin(2π/3) = 32√3 B1; difference B1
  3. (c) 16π/3 + 16 cm
    arc AC = 8 × 2π/3 = 16π/3 B1; AB + BC = 16 B1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Arc length and sector area Banker9 questions · 9/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Arc length and sector area Banker689/9

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