Circular Measure Short Questions (Question 1, 2 & 3)


Question 1:

The figure shows the sector OCB of radius 13 cm at the centre O. The length of the arc CB = 5.2 cm. Find
(a) the angle in radians,
(b) the perimeter of the shaded region.

Solution:
(a)
s=rθ 5.2=13( COB ) COB=0.4 radian

(b)
cosCOB= OA OC cos0.4= OA 13  (change calculator to Rad mode) OA=11.97 cm AB=1311.97=1.03 cm CA= 13 2 11.97 2 CA=5.07 cm

Perimeter of the shaded region = 5.07 + 1.03 + 5.2 = 11.3 cm.



Question 2:
The figure shows the sector AOB of a circle, centre O and radius 5 cm. The length of the arc AB is 6 cm. Find the area of:
(a) the sector AOB,
(b) the shaded region.

Solution:
(a) Arc AB = 6cm
      s = θ
      6 = 5 θ
        θ = 6/5 rad

Area of sector AOB = 1 2 r 2 θ= 1 2 ( 5 ) 2 ( 6 5 )=15c m 2

(b)
Area of shaded region = 1 2 r 2 ( θsinθ ) ( change calculator to Rad mode ) = 1 2 ( 5 ) 2 ( 6 5 sin 6 5 ) =3.35  cm 2


Question 3:
Diagram below shows a sector QOR of a circle with centre O.

It is given that PS = 8 cm and QP = PO = OS = SR = 5 cm.
Find
(a) the length, in cm, of the arc QR,
(b) the area, in cm2, of the shaded region.

Solution:
(a) Length of arc QR = θ = 10 (1.75) = 17.5 cm

(b)
Area of the shaded region
= Area of sector QOR – Area of triangle POS
½ (10)2 (1.75) – ½ (5) (5) sin 1.75 (change calculator to Rad mode)
= 87.5 – 12.30
= 75.2 cm2