37 exam-style questions on The circle: arc length; sector area — topic 3.4 of Geometry and Trigonometry in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
A circle has radius 8 cm. Calculate the length of the arc that subtends an angle of 45° at the centre. Give your answer to 3 s.f.
Answer: 6.28 cm
Arc length = (θ/360)×2πr = (45/360)×2π×8 = π×2 = 6.28 cm. Using the whole circumference (50.3 cm) or the sector area formula are common errors.
The area of a sector is 30 cm² and its radius is 5 cm. Determine the central angle of the sector in degrees, to 3 s.f.
Answer: 138°
Area = (θ/360)×πr² ⇒ 30 = (θ/360)×π×25 ⇒ θ = 30×360/(25π) = 137.5° ≈ 138° (3 s.f.).
A circular pizza of radius 15 cm is cut into 8 equal slices. Calculate the area of one slice to 3 s.f.
Answer: 88.4 cm²
One slice = 1/8 of the whole area = (1/8)×π×15² = (1/8)×225π = 88.4 cm². 11.8 cm² is the arc length, not the area.
A goat is tethered by a 7 m rope to a corner of a large field, and can graze over a quarter-circle. Calculate the grazing area to 3 s.f.
Answer: 38.5 m²
A quarter circle has area (1/4)×πr² = (1/4)×π×49 = 38.5 m². 11.0 m is the arc length; 154 m² is the full circle.
A windscreen wiper of length 45 cm sweeps through an angle of 110°. Calculate the area of the region it cleans, to 3 s.f.
Answer: 1940 cm²
Swept area = (110/360)×π×45² = (110/360)×2025π = 1940 cm². 86.4 cm is the arc length at the tip; 6360 cm² is the full circle.
Determine the area of a sector whose arc length is 9 cm and whose radius is 6 cm. (Hint: sector area = (1/2)×arc×radius for consistent measure.)
Answer: 27 cm²
Sector area = (1/2)×arc×r = (1/2)×9×6 = 27 cm². This shortcut avoids finding the angle.
That was 6 of 37 The circle: arc length; sector area questions.
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37 The circle: arc length; sector area questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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