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AS & A Level · Practice questions · Arc length and sector area

Slice the circle.

Six original Cambridge-style questions on circular measure: arc length, sector area, segments, tangents and a compound region in context.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

Name the pieces, then use \(r\theta\) and \(\tfrac{1}{2}r^2\theta\).

01
Recall
[2 marks]

A sector has radius 9 cm and angle \(\dfrac{2\pi}{9}\) radians. Find the exact arc length and the exact area.

  • Arc \(= 9 \times \dfrac{2\pi}{9} = 2\pi\) cm ✓
  • Area \(= \tfrac{1}{2} \times 81 \times \dfrac{2\pi}{9} = 9\pi\ \text{cm}^2\) ✓
02
Calculation
[3 marks]

A sector of a circle of radius 12 cm has a perimeter of 40 cm. Find the angle of the sector and its area.

  • Arc \(= 40 - 24 = 16\), so \(\theta = \dfrac{16}{12} = \dfrac{4}{3}\) radians. ✓
  • Area \(= \tfrac{1}{2} \times 144 \times \dfrac{4}{3} = 96\ \text{cm}^2\) ✓
03
Calculation
[4 marks]

A chord \(PQ\) of a circle with centre \(O\) and radius 10 cm subtends an angle of 2.2 radians at \(O\). Find the area and the perimeter of the minor segment cut off by \(PQ\).

  • Area \(= \tfrac{1}{2} \times 100 \times (2.2 - \sin 2.2) = 50(2.2 - 0.80850) = 69.6\ \text{cm}^2\) ✓
  • Arc \(= 10 \times 2.2 = 22\); chord \(= 2 \times 10\sin 1.1 = 17.82\) ✓
  • Perimeter \(= 22 + 17.82 = 39.8\) cm ✓
04
Proof
[4 marks]

A sector \(OAB\) has radius \(r\) cm and a perimeter of \(4r\) cm.

(a) Show that the angle \(AOB\) is 2 radians.

(b) Show that the area of the minor segment cut off by the chord \(AB\) is \(\tfrac{1}{2}r^2(2 - \sin 2)\), and find its value when \(r = 5\).

  • (a) \(2r + r\theta = 4r\), so \(r\theta = 2r\) and \(\theta = 2\). ✓
  • (b) Segment = sector \(-\) triangle \(= \tfrac{1}{2}r^2 \times 2 - \tfrac{1}{2}r^2\sin 2 = \tfrac{1}{2}r^2(2 - \sin 2)\). ✓
  • When \(r = 5\): \(12.5 \times (2 - 0.90930) = 13.6\ \text{cm}^2\) ✓
05
Analysis
[6 marks]

Sector \(OAB\) has centre \(O\), radius 10 cm and angle \(AOB = \dfrac{\pi}{6}\). The tangent to the arc at \(A\) meets the line \(OB\) extended at \(C\).

(a) Find the exact lengths of \(AC\) and \(OC\).

(b) Find the exact perimeter of the region bounded by the arc \(AB\) and the lines \(BC\) and \(CA\).

(c) Find the exact area of this region.

  • (a) Angle \(OAC = \dfrac{\pi}{2}\). \(AC = 10\tan\dfrac{\pi}{6} = \dfrac{10}{\sqrt{3}} = \dfrac{10\sqrt{3}}{3}\); \(OC = \dfrac{10}{\cos(\pi/6)} = \dfrac{20}{\sqrt{3}} = \dfrac{20\sqrt{3}}{3}\). ✓
  • (b) Arc \(= 10 \times \dfrac{\pi}{6} = \dfrac{5\pi}{3}\); \(BC = \dfrac{20\sqrt{3}}{3} - 10\). ✓
  • Perimeter \(= \dfrac{5\pi}{3} + \dfrac{20\sqrt{3}}{3} - 10 + \dfrac{10\sqrt{3}}{3} = \dfrac{5\pi}{3} + 10\sqrt{3} - 10\) cm ✓
  • (c) Triangle \(OAC = \tfrac{1}{2} \times 10 \times \dfrac{10\sqrt{3}}{3} = \dfrac{50\sqrt{3}}{3}\); sector \(= \tfrac{1}{2} \times 100 \times \dfrac{\pi}{6} = \dfrac{25\pi}{3}\). ✓
  • Area \(= \dfrac{50\sqrt{3}}{3} - \dfrac{25\pi}{3}\ \text{cm}^2\) ✓
06
Analysis
[6 marks]

A windscreen wiper arm \(OP\) of length 50 cm rotates about \(O\) through an angle of 2.2 radians. The rubber blade lies along the arm from 15 cm to 50 cm from \(O\).

(a) Find the area of windscreen wiped by the blade.

(b) Find the perimeter of the wiped region.

(c) Find the straight-line distance between the positions of the tip \(P\) at the two ends of its sweep.

  • (a) The wiped region is a large sector minus a small sector: \(\tfrac{1}{2} \times 2.2 \times (50^2 - 15^2) = 1.1 \times 2275 = 2502.5\), so \(2500\ \text{cm}^2\) (3 s.f.). ✓
  • (b) Outer arc \(= 50 \times 2.2 = 110\); inner arc \(= 15 \times 2.2 = 33\); two straight edges of \(50 - 15 = 35\). ✓
  • Perimeter \(= 110 + 33 + 35 + 35 = 213\) cm ✓
  • (c) The chord is \(2 \times 50\sin 1.1 = 100 \times 0.89121 = 89.1\) cm ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.