CAIE A-Level Mathematics 1.4.2 Arc Length and Sector Area

CAIE A-Level Mathematics 1.4.2 Arc Length and Sector Area
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise applying s = rθ and A = ½r²θ to sectors and compound figures, using triangle geometry to obtain unknown angles, radii, perimeters and segment areas.

How this is tested

  • calculate arc length with s = rθ and add radial or chord lengths for a perimeter
  • calculate sector area with A = ½r²θ and subtract or add the relevant triangle area
  • derive an unknown r or θ from area or perimeter data before giving an exact final result

Question 7(a)

[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 cmr \mathrm{~cm}. The shaded region is bounded by the chord AB and the arc⁡AB\operatorname{arc} A B. The size of angle AOB is 23π\frac{2}{3} \pi radians.

Show that the area of the shaded region is approximately 0.614r2 cm20.614 r^{2} \mathrm{~cm}^{2}.

It is given that the radius of the circle is increasing at a rate of 0.4 cm s−10.4 \mathrm{~cm} \mathrm{~s}^{-1}.