Question Simulator

AS & A Level Mathematics (9709)

Circular measure Banker

31 questions · about 2% of the marks we indexed · examined in 9 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 11 · Q7 (a, b)8 marksArc length & sector area
  • 2026 May/June · Paper 13 · Q7 (a, b)6 marksArc length & sector area
  • 2026 May/June · Paper 15 · Q8 (a)2 marksArc length & sector area
  • 2026 May/June · Paper 15 · Q11 (b)5 marksArc length & sector area

2025

  • 2025 Oct/Nov · Paper 11 · Q7 (a)2 marksArc length & sector area
  • 2025 Oct/Nov · Paper 12 · Q10 (a, b)9 marksArc length & sector area
  • 2025 Oct/Nov · Paper 13 · Q5 (a, b)5 marksArc length & sector area
  • 2025 Oct/Nov · Paper 15 · Q4 (a, b)7 marksArc length & sector area
  • 2025 May/June · Paper 11 · Q9 (a, b)8 marksArc length & sector area
  • 2025 May/June · Paper 12 · Q8 (b, c)6 marksArc length & sector area
  • 2025 May/June · Paper 13 · Q8 (a, b, c)8 marksArc length & sector area
  • 2025 May/June · Paper 15 · Q5 (a, b)6 marksArc length & sector area

2024

  • 2024 Oct/Nov · Paper 11 · Q35 marksArc length & sector area
  • 2024 Oct/Nov · Paper 12 · Q66 marksArc length & sector area
  • 2024 Oct/Nov · Paper 13 · Q7 (a, b)10 marksArc length & sector area
  • 2024 May/June · Paper 11 · Q7 (a, b)8 marksOfficial paper ↗Arc length & sector area
  • 2024 May/June · Paper 12 · Q8 (a(i), a(ii), b)10 marksArc length & sector area
  • 2024 May/June · Paper 13 · Q3 (a, b)6 marksArc length & sector area

2023

  • 2023 Oct/Nov · Paper 11 · Q6 (a, b, c)7 marksArc length & sector area
  • 2023 Oct/Nov · Paper 12 · Q4 (a, b)6 marksArc length & sector area
  • 2023 Oct/Nov · Paper 13 · Q10 (a, b, c)9 marksArc length & sector area
  • 2023 May/June · Paper 11 · Q44 marksArc length & sector area
  • 2023 May/June · Paper 12 · Q6 (a, b)6 marksArc length & sector area
  • 2023 May/June · Paper 13 · Q6 (a, b)7 marksArc length & sector area

2022

  • 2022 Oct/Nov · Paper 11 · Q5 (a, b)6 marksArc length & sector area
  • 2022 Oct/Nov · Paper 12 · Q10 (a, b, c)8 marksArc length & sector area
  • 2022 Oct/Nov · Paper 13 · Q8 (a, b)7 marksArc length & sector area
  • 2022 Oct/Nov · Paper 13 · Q10 (c)2 marksArc length & sector area
  • 2022 May/June · Paper 11 · Q5 (a, b)7 marksArc length & sector area
  • 2022 May/June · Paper 12 · Q7 (a, b)6 marksArc length & sector area
  • 2022 May/June · Paper 13 · Q9 (a, b)8 marksArc length & 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 & sector area · Core · Paper 1 style · 4 marks
  1. (a) A sector OAB of a circle has centre O, radius 12 cm and angle AOB = 0.85 radians. Find the length of the arc AB. [2]
  2. (b) Find the area of the sector OAB. [2]
Show answer
  1. (a) 10.2 cm
    s = rθ = 12 × 0.85 M1; = 10.2 cm A1
  2. (b) 61.2 cm²
    A = ½r²θ = ½ × 144 × 0.85 M1; = 61.2 cm² A1
Q2Arc length & sector area · Core · Paper 1 style · 4 marks
  1. (a) Express 135° in radians, giving your answer as an exact multiple of π. [1]
  2. (b) A sector of a circle of radius 8 cm has angle 135°. Find, in terms of π, the perimeter and the area of the sector. [3]
Show answer
  1. (a) 3π/4
    135 × π/180 = 3π/4 B1
  2. (b) Perimeter = 16 + 6π cm; area = 24π cm²
    Arc = 8 × 3π/4 = 6π B1; perimeter = 8 + 8 + 6π = 16 + 6π B1; area = ½ × 64 × 3π/4 = 24π B1
Q3Arc length & sector area · Core · Paper 1 style · 3 marks
  1. (a) A car windscreen wiper arm is pivoted at a point O. The rubber blade lies along the arm from 15 cm to 50 cm from O. In one sweep the arm turns through an angle of 2.1 radians. Find the area of windscreen cleaned in one sweep. Give your answer correct to 3 significant figures. [3]
Show answer
  1. (a) 2390 cm²
    Area = ½ × 50² × 2.1 − ½ × 15² × 2.1 M1 M1 (difference of two sectors); = 2388.75 ≈ 2390 cm² A1
Q4Arc length & sector area · Core · Paper 1 style · 4 marks
  1. (a) A sector of a circle has radius 6 cm and area 45 cm². Find the angle of the sector in radians. [2]
  2. (b) Find the perimeter of the sector. [2]
Show answer
  1. (a) 2.5 radians
    ½ × 36 × θ = 45 M1; θ = 2.5 A1
  2. (b) 27 cm
    Arc = 6 × 2.5 = 15 M1; perimeter = 15 + 12 = 27 cm A1
Q5Arc length & sector area · Core · Paper 1 style · 4 marks
  1. (a) A sector of a circle has arc length 14 cm and area 70 cm². Find the radius of the circle and the angle of the sector in radians. [4]
Show answer
  1. (a) r = 10 cm, θ = 1.4 radians
    rθ = 14 and ½r²θ = 70 B1; ½r(rθ) = 70 ⇒ 7r = 70 M1; r = 10 A1; θ = 14/10 = 1.4 A1
Q6Arc length & sector area · Core · Paper 1 style · 5 marks
  1. (a) A round pizza of diameter 30 cm is cut into slices from its centre. One slice is a sector with angle π/5 radians. Find, in terms of π, the length of the crust (the curved edge) of this slice. [1]
  2. (b) Find, in terms of π, the area of the top surface of this slice. [2]
  3. (c) Ayesha cuts a second slice whose top surface area is 18π cm². Find the angle of this slice in radians. [2]
Show answer
  1. (a) 3π cm
    s = 15 × π/5 = 3π B1
  2. (b) 45π/2 cm² (= 22.5π cm²)
    ½ × 15² × π/5 M1; = 45π/2 A1
  3. (c) 4π/25 radians
    ½ × 225 × θ = 18π M1; θ = 36π/225 = 4π/25 A1
Q7Arc length & sector area · Core · Paper 1 style · 4 marks
  1. (a) A pendulum consists of a small bob on a light rod of length 80 cm. The bob swings from 0.3 radians on one side of the vertical to 0.3 radians on the other side. Find the distance travelled by the bob in one swing from one extreme position to the other. [2]
  2. (b) Find how much higher the bob is at an extreme position than at its lowest point. Give your answer correct to 3 significant figures. [2]
Show answer
  1. (a) 48 cm
    Angle turned = 0.6 rad B1; s = 80 × 0.6 = 48 cm B1
  2. (b) 3.57 cm
    Height = 80 − 80 cos 0.3 M1; = 3.57 cm A1
Q8Arc length & sector area · Core · Paper 1 style · 5 marks
  1. (a) The minute hand of the clock on a clock tower in Faisalabad is 1.8 m long. Find the angle, in radians, that the minute hand turns through in 20 minutes. [1]
  2. (b) Find the distance moved by the tip of the minute hand in 20 minutes, correct to 3 significant figures. [2]
  3. (c) Find the area swept out by the minute hand in 20 minutes, correct to 3 significant figures. [2]
Show answer
  1. (a) 2π/3 radians
    20/60 × 2π = 2π/3 B1
  2. (b) 3.77 m
    s = 1.8 × 2π/3 = 1.2π M1; = 3.77 m A1
  3. (c) 3.39 m²
    ½ × 1.8² × 2π/3 = 1.08π M1; = 3.39 m² A1
Q9Arc length & sector area · Core · Paper 1 style · 5 marks
  1. (a) Points A and B lie on a circle with centre O and radius 5 cm, and angle AOB = 1.2 radians. Find the length of the chord AB. [2]
  2. (b) Find the area of the minor segment cut off by the chord AB. [3]
Show answer
  1. (a) 5.65 cm
    AB = 2 × 5 sin 0.6 (or cosine rule) M1; = 5.65 cm A1
  2. (b) 3.35 cm²
    Sector = ½ × 25 × 1.2 = 15 B1; triangle = ½ × 25 sin 1.2 = 11.65 B1; segment = 15 − 11.65 = 3.35 cm² A1
Q10Arc length & sector area · Core · Paper 1 style · 3 marks
  1. (a) A sector of a circle has angle 2 radians and perimeter 28 cm. Find the radius of the circle. [2]
  2. (b) Find the area of the sector. [1]
Show answer
  1. (a) 7 cm
    2r + 2r = 28 M1; r = 7 A1
  2. (b) 49 cm²
    ½ × 49 × 2 = 49 B1
Q11Arc length & sector area · Standard · Paper 1 style · 5 marks
  1. (a) A sector OAB has centre O, radius 10 cm and angle AOB = θ radians. The perimeter of the sector is 32 cm. Find θ. [2]
  2. (b) Find the area of the region bounded by the chord AB and the arc AB, correct to 3 significant figures. [3]
Show answer
  1. (a) θ = 1.2
    20 + 10θ = 32 M1; θ = 1.2 A1
  2. (b) 13.4 cm²
    Sector = ½ × 100 × 1.2 = 60 M1; triangle = ½ × 100 sin 1.2 = 46.60 M1; 60 − 46.60 = 13.4 cm² A1
Q12Arc length & sector area · Standard · Paper 1 style · 5 marks
  1. (a) In a park in Islamabad, a curved path is the region between two arcs with the same centre O. The inner arc has radius 7 m and the outer arc has radius 9 m, and both arcs subtend an angle of 1.5 radians at O. The two straight ends of the path lie along radii. Find the area of the path. [2]
  2. (b) Find the total length of the boundary of the path. [2]
  3. (c) Edging is fixed along the whole boundary at a cost of Rs 850 per metre. Find the total cost. [1]
Show answer
  1. (a) 24 m²
    ½ × 9² × 1.5 − ½ × 7² × 1.5 M1; = 60.75 − 36.75 = 24 m² A1
  2. (b) 28 m
    9 × 1.5 + 7 × 1.5 + 2 + 2 M1; = 28 m A1
  3. (c) Rs 23 800
    28 × 850 = 23 800 B1
Q13Arc length & sector area · Standard · Paper 1 style · 5 marks
  1. (a) Points A and B lie on a circle with centre O and radius 8 cm. The chord AB has length 10 cm. Show that angle AOB = 1.350 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) angle AOB = 1.350 rad
    sin(½AOB) = 5/8 M1; AOB = 2 sin⁻¹(0.625) = 1.3503 A1 (or cosine rule: cos AOB = (64 + 64 − 100)/128)
  2. (b) 12.0 cm²
    Sector ½ × 64 × 1.3503 = 43.21 M1; triangle ½ × 64 sin 1.3503 = 31.22 M1; 11.99 ≈ 12.0 cm² A1
Q14Arc length & sector area · Standard · Paper 1 style · 6 marks
  1. (a) The sector OAB has centre O, radius 5 cm and angle AOB = 0.9 radians. The tangent to the arc at A meets the line OB extended at T. Find the area of the region bounded by AT, TB and the arc AB, correct to 3 significant figures. [3]
  2. (b) Find the perimeter of this region, correct to 3 significant figures. [3]
Show answer
  1. (a) 4.50 cm²
    AT = 5 tan 0.9 = 6.301 M1; triangle OAT = ½ × 5 × 6.301 = 15.75, sector = 11.25 M1; region = 4.50 cm² A1
  2. (b) 13.8 cm
    OT = 5/cos 0.9 = 8.044, so BT = 3.044 B1; arc = 4.5 B1; perimeter = 6.301 + 3.044 + 4.5 = 13.8 cm A1
Q15Arc length & sector area · Standard · Paper 1 style · 6 marks
  1. (a) A badge is made from an equilateral triangle ABC of side 4 cm. Each side is replaced by an arc of a circle centred at the opposite vertex (for example, arc BC has centre A and radius 4 cm). Find, in terms of π, the perimeter of the badge. [2]
  2. (b) Show that the area of the badge is 8(π − √3) cm². [4]
Show answer
  1. (a) 4π cm
    Each arc = 4 × π/3 M1; total = 3 × 4π/3 = 4π A1
  2. (b) 8(π − √3) cm²
    Triangle area = ½ × 16 × sin(π/3) = 4√3 B1; each segment = ½ × 16 × π/3 − 4√3 = 8π/3 − 4√3 M1 A1; total = 4√3 + 3(8π/3 − 4√3) = 8π − 8√3 A1
Q16Arc length & sector area · Standard · Paper 1 style · 6 marks
  1. (a) A chord AB of a circle with centre O and radius 6 cm subtends an angle of 2π/3 radians at O. Find the exact area of the minor segment cut off by AB. [3]
  2. (b) Find the exact perimeter of the minor segment. [3]
Show answer
  1. (a) 12π − 9√3 cm²
    Sector = ½ × 36 × 2π/3 = 12π B1; triangle = ½ × 36 × sin(2π/3) = 9√3 B1; segment = 12π − 9√3 A1
  2. (b) 4π + 6√3 cm
    Arc = 6 × 2π/3 = 4π B1; AB = 2 × 6 sin(π/3) = 6√3 M1; perimeter = 4π + 6√3 A1
Q17Arc length & sector area · Standard · Paper 1 style · 6 marks
  1. (a) OABC is a rhombus with sides of length 7 cm and angle AOC = 1.1 radians. An arc AC of a circle with centre O and radius 7 cm is drawn inside the rhombus. Find the area of the region inside the rhombus but outside the sector OAC, correct to 3 significant figures. [4]
  2. (b) Find the perimeter of this region. [2]
Show answer
  1. (a) 16.7 cm²
    Rhombus area = 7 × 7 × sin 1.1 = 43.67 M1 A1; sector = ½ × 49 × 1.1 = 26.95 M1; 43.67 − 26.95 = 16.7 cm² A1
  2. (b) 21.7 cm
    AB + BC + arc AC = 7 + 7 + 7 × 1.1 M1; = 21.7 cm A1
Q18Arc length & sector area · Standard · Paper 1 style · 5 marks
  1. (a) A sector of a circle has radius r cm and angle θ radians. The perimeter of the sector is equal to twice the length of its arc. Find θ. [2]
  2. (b) The area of the sector is 36 cm². Find r and the perimeter of the sector. [3]
Show answer
  1. (a) θ = 2
    2r + rθ = 2rθ M1; θ = 2 A1
  2. (b) r = 6 cm, perimeter = 24 cm
    ½r² × 2 = 36 M1; r = 6 A1; perimeter = 12 + 12 = 24 cm A1
Q19Arc length & sector area · Standard · Paper 1 style · 5 marks
  1. (a) A circular garden in the middle of a roundabout in Lahore has centre O and radius 15 m. A straight footpath crosses the garden along a chord AB of length 24 m. Find angle AOB in radians, correct to 4 significant figures. [2]
  2. (b) Find the area of the smaller of the two parts into which the path divides the garden, correct to 3 significant figures. [3]
Show answer
  1. (a) 1.855 radians
    sin(½AOB) = 12/15 M1; AOB = 2 sin⁻¹ 0.8 = 1.855 A1
  2. (b) 101 m²
    Sector = ½ × 225 × 1.8546 = 208.6 M1; triangle = ½ × 24 × 9 = 108 B1; 208.6 − 108 = 101 m² A1
Q20Arc length & sector area · Standard · Paper 1 style · 6 marks
  1. (a) The sector OAB has centre O, radius 9 cm and angle AOB = 1.2 radians. The point C lies on OA with OC = 5 cm. Find the area of the region R bounded by the line segments CA and CB and the arc AB, correct to 3 significant figures. [3]
  2. (b) Find the perimeter of R, correct to 3 significant figures. [3]
Show answer
  1. (a) 27.6 cm²
    Sector = ½ × 81 × 1.2 = 48.6 B1; triangle OCB = ½ × 5 × 9 × sin 1.2 = 20.97 M1; R = 27.6 cm² A1
  2. (b) 23.4 cm
    BC² = 25 + 81 − 90 cos 1.2 M1; BC = 8.567 A1; perimeter = 4 + 10.8 + 8.567 = 23.4 cm A1
Q21Arc length & sector area · Challenge · Paper 1 style · 8 marks
  1. (a) Two circles intersect at A and B. The first has centre P and radius 6 cm; the second has centre Q and radius 8 cm, and PQ = 10 cm. Show that angle PAQ is a right angle, and find angles APB and AQB in radians, correct to 4 significant figures. [4]
  2. (b) Find the area of the region common to both circles, correct to 3 significant figures. [4]
Show answer
  1. (a) APB = 1.855 rad, AQB = 1.287 rad
    6² + 8² = 10² so PAQ = 90° (converse of Pythagoras) B1; cos APQ = 6/10 M1; APB = 2 cos⁻¹ 0.6 = 1.855 A1; AQB = 2 cos⁻¹ 0.8 = 1.287 A1
  2. (b) 26.6 cm²
    Segment in circle P: ½ × 36(1.8546 − sin 1.8546) = 16.10 M1 A1; segment in circle Q: ½ × 64(1.2870 − sin 1.2870) = 10.46 M1; total = 26.6 cm² A1
Q22Arc length & sector area · Challenge · Paper 1 style · 6 marks
  1. (a) A farmer in Sindh uses 24 m of fencing to enclose a sector-shaped pen of radius r m and angle θ radians. The fencing goes along both radii and the arc. Show that the area A m² of the pen is given by A = 12r − r². [3]
  2. (b) Find the maximum area of the pen and the value of θ for which it occurs. [3]
Show answer
  1. (a) A = 12r − r²
    2r + rθ = 24 ⇒ θ = (24 − 2r)/r M1; A = ½r²θ = ½r(24 − 2r) M1; = 12r − r² A1
  2. (b) Maximum area 36 m² when r = 6 and θ = 2
    dA/dr = 12 − 2r = 0 (or complete the square: 36 − (r − 6)²) M1; r = 6, A = 36 A1; θ = (24 − 12)/6 = 2 A1
Q23Arc length & sector area · Challenge · Paper 1 style · 5 marks
  1. (a) Three pipes, each with a circular cross-section of radius 5 cm, are tied together so that each pipe touches the other two. Find the exact area of the cross-section of the gap enclosed between the three pipes. [4]
  2. (b) Hence give this area correct to 3 significant figures. [1]
Show answer
  1. (a) 25√3 − 25π/2 cm²
    Centres form an equilateral triangle of side 10 B1; triangle area = ½ × 100 × sin(π/3) = 25√3 M1; three sectors of angle π/3: 3 × ½ × 25 × π/3 = 25π/2 M1; gap = 25√3 − 25π/2 A1
  2. (b) 4.03 cm²
    25√3 − 25π/2 = 4.03 B1
Q24Arc length & sector area · Challenge · Paper 1 style · 6 marks
  1. (a) A sector OAB has centre O, radius 12 cm and angle AOB = π/3. A circle with centre C and radius r cm lies inside the sector and touches OA, OB and the arc AB. Show that OC = 2r and hence find r. [4]
  2. (b) Find, in terms of π, the area of the part of the sector that lies outside the circle. [2]
Show answer
  1. (a) OC = 2r, r = 4
    C lies on the bisector of angle AOB, so the angle between OC and OA is π/6 B1; r = OC sin(π/6) so OC = 2r M1 A1; OC + r = 12 ⇒ 3r = 12, r = 4 A1
  2. (b) 8π cm²
    Sector = ½ × 144 × π/3 = 24π; circle = 16π M1; 24π − 16π = 8π A1
Q25Arc length & sector area · Challenge · Paper 1 style · 7 marks
  1. (a) Triangle ABC is isosceles with AB = AC = 10 cm and angle BAC = 0.8 radians. An arc BC is drawn with centre A and radius 10 cm. A semicircle is drawn on BC as diameter, on the side of BC away from A. Find the length of BC, correct to 4 significant figures. [2]
  2. (b) Find the area of the region inside the semicircle but outside the sector ABC, correct to 3 significant figures. [5]
Show answer
  1. (a) BC = 7.788 cm
    BC = 2 × 10 sin 0.4 M1; = 7.788 cm A1
  2. (b) 19.7 cm²
    Semicircle = ½π(3.894)² = 23.82 M1 A1; segment of sector = ½ × 100(0.8 − sin 0.8) = 4.132 M1 A1; region = 23.82 − 4.13 = 19.7 cm² A1
Q26Arc length & sector area · Challenge · Paper 1 style · 7 marks
  1. (a) ABCD is a square of side 8 cm. Two quarter circles of radius 8 cm are drawn inside the square, one with centre A passing through B and D, and one with centre C passing through B and D. Find, in terms of π, the perimeter of the leaf-shaped region that lies inside both quarter circles. [2]
  2. (b) Show that the area of the leaf-shaped region is 32(π − 2) cm². [3]
  3. (c) Find the percentage of the square that lies outside the leaf, correct to 3 significant figures. [2]
Show answer
  1. (a) 8π cm
    Each arc = 8 × π/2 = 4π M1; perimeter = 8π A1
  2. (b) 32(π − 2) cm²
    Half the leaf = quarter circle − triangle ABD = 16π − 32 M1 A1; leaf = 2(16π − 32) = 32π − 64 = 32(π − 2) A1
  3. (c) 42.9%
    (64 − 36.53)/64 × 100 M1; = 42.9% A1
Q27Arc length & sector area · Challenge · Paper 1 style · 8 marks
  1. (a) In a rice mill, a tight belt passes around two pulleys with radii 9 cm and 4 cm. The centres of the pulleys are 13 cm apart. The straight parts of the belt are common external tangents to the two circles. Show that each straight part has length 12 cm. [2]
  2. (b) Let α = sin⁻¹(5/13). Explain why the belt is in contact with the larger pulley over an angle of (π + 2α) radians, and find α correct to 4 significant figures. [3]
  3. (c) Find the total length of the belt, correct to 3 significant figures. [3]
Show answer
  1. (a) 12 cm
    Tangent length = √(13² − (9 − 4)²) M1; = √144 = 12 A1
  2. (b) α = 0.3948 rad
    Radii to the tangent points are perpendicular to the straight parts, which make angle α with the line of centres where sin α = 5/13 M1; so the contact angle on the larger pulley is π + 2α (and on the smaller π − 2α) A1; α = 0.3948 B1
  3. (c) 68.8 cm
    L = 2 × 12 + 9(π + 2α) + 4(π − 2α) M1; = 24 + 13π + 10α M1; = 68.8 cm A1
Q28Arc length & sector area · Challenge · Paper 1 style · 7 marks
  1. (a) A street vendor in Karachi makes paper cones for roasted chana. Each cone is made by joining the two straight edges of a sector of paper of radius 15 cm and angle θ radians, without overlap. The circular rim of the cone has radius 6 cm. Find θ, giving your answer as an exact multiple of π. [2]
  2. (b) Find, in terms of π, the area of paper used for each cone. [2]
  3. (c) Find the volume of chana a full cone can hold, correct to 3 significant figures. [3]
Show answer
  1. (a) θ = 4π/5
    Arc length = circumference of rim: 15θ = 2π × 6 M1; θ = 12π/15 = 4π/5 A1
  2. (b) 90π cm²
    ½ × 225 × 4π/5 M1; = 90π A1
  3. (c) 518 cm³
    Height = √(15² − 6²) = √189 = 13.75 M1 A1; V = ⅓π × 36 × 13.75 = 518 cm³ A1
Q29Arc length & sector area · Challenge · Paper 1 style · 7 marks
  1. (a) A circle has centre O and radius 5 cm. A point T is 13 cm from O, and the tangents from T touch the circle at A and B. Show that angle AOB = 2.352 radians, correct to 4 significant figures. [2]
  2. (b) Find the area of the region bounded by TA, TB and the minor arc AB, correct to 3 significant figures. [3]
  3. (c) Find the perimeter of this region, correct to 3 significant figures. [2]
Show answer
  1. (a) AOB = 2.352 rad
    OA ⟂ TA, so cos AOT = 5/13 M1; AOB = 2 cos⁻¹(5/13) = 2.3520 A1
  2. (b) 30.6 cm²
    TA = √(13² − 5²) = 12 B1; kite OATB = 2 × ½ × 5 × 12 = 60, sector = ½ × 25 × 2.3520 = 29.40 M1; region = 60 − 29.40 = 30.6 cm² A1
  3. (c) 35.8 cm
    12 + 12 + 5 × 2.3520 M1; = 35.8 cm A1
Q30Arc length & sector area · Challenge · Paper 1 style · 7 marks
  1. (a) A window in a Mughal-style building is a rectangle 1.2 m wide and 1.5 m tall, topped by a pointed arch. The arch is formed by two arcs, each of radius 1.2 m. One arc has its centre at the top left corner of the rectangle and the other at the top right corner; the arcs meet at the top point P of the window. Explain why the centres of the arcs and P form an equilateral triangle, and show that the area of the arch region above the rectangle is (0.48π − 0.36√3) m². [4]
  2. (b) Find the total area of the window, correct to 3 significant figures. [1]
  3. (c) Find the perimeter of the window, correct to 3 significant figures. [2]
Show answer
  1. (a) (0.48π − 0.36√3) m²
    Each centre is 1.2 m from P and the centres are 1.2 m apart, so all sides are 1.2 m B1; each arc subtends π/3 at its centre B1; arch = 2 × ½ × 1.44 × π/3 − ½ × 1.44 × sin(π/3) M1; = 0.48π − 0.36√3 A1
  2. (b) 2.68 m²
    1.8 + 0.884 = 2.68 m² B1
  3. (c) 6.71 m
    1.2 + 2 × 1.5 + 2 × 1.2 × π/3 M1; = 6.71 m A1

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