CAIE A-Level Mathematics 1.4 Circular Measure Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise working in radians and combining arc length, sector area, chord and triangle geometry to solve perimeters, segments, shaded regions and changing-area problems.

Exam points

use s = rθ and A = ½r²θ only after expressing the central angle in radians
combine arc lengths with straight sides and sector areas with triangle areas
solve simultaneous perimeter or area conditions for r and θ before selecting valid values

Covered syllabus points

1.4.1—Radians 1.4.2—Arc length and sector area

[Maximum number: 7]

Figure for Question 6 — CAIE A-Level Mathematics

The diagram shows a motif formed by the major arc A B of a circle with radius r and centre O, and the minor $\operatorname{arc} A O B$ of a circle, also with radius r but with centre C. The point C lies on the circle with centre O.

(a)

Given that angle $A C B=k \pi$ radians, state the value of the fraction k.

[ 1 ]

(b)

State the perimeter of the shaded motif in terms of $\pi$ and r.

[ 1 ]

(c)

Find the area of the shaded motif, giving your answer in terms of $\pi, r$ and $\sqrt{3}$ .

[ 5 ]

[Maximum number: 2]

Figure for Question 7(a) — CAIE A-Level Mathematics

The diagram shows a sector of a circle with centre O and radius $r \mathrm{~cm}$ . The shaded region is bounded by the chord AB and the $\operatorname{arc} A B$ . The size of angle AOB is $\frac{2}{3} \pi$ radians.

Show that the area of the shaded region is approximately $0.614 r^{2} \mathrm{~cm}^{2}$ .

It is given that the radius of the circle is increasing at a rate of $0.4 \mathrm{~cm} \mathrm{~s}^{-1}$ .

Exam points

Covered syllabus points

Question 6

Question 6(a)

Question 6(b)

Question 6(c)

Question 7(a)