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1.4.2 Circular Measure Short Questions (Question 4, 5 & 6)


Question 4:
Diagram below shows a circle with centre O.
The length of the minor arc is 16 cm and the angle of the major sector AOB is 290o.
Using  π = 3.142, find
(a) the value of  θ, in radians. (Give your answer correct to four significant figures)
(b) the length, in cm, of the radius of the circle.

Solution:
(a) 
Angle of the minor sector AOB
= 360o – 290o
= 70o
= 70o × 3.142 180
= 1.222 radians

(b) 
Using s = rθ
r × 1.222 = 16
radius, r = 13.09 cm


Question 5:
Diagram below shows sector OPQ with centre O and sector PXY with centre P.
Given that OQ = 8 cm, PY = 3 cm ,  ∠ XPY = 1.2 radians and the length of arc PQ = 6cm ,
calculate
( a)  the value of θ , in  radian ,
( b)  the area, in cm2 , of the shaded region .

Solution:
(a) s = r θ
 6 = 8 θ
 θ = 0.75 rad

(b) 
Area of the shaded region
= Area of sector OPQ – Area of sector PXY
= 1 2 ( 8 ) 2 ( 0.75 ) − 1 2 ( 3 ) 2 ( 1.2 )
= 24 – 5.4
= 18.6 cm2



Question 6:
Diagram below shows a circle with centre O and radius 12 cm.

Given that A, B and C are points such that OA = AB and  ∠OAC = 90°, find
(a)   ∠BOC, in radians,
(b)  the area, in cm2, of the shaded region.   

Solution:
(a) For triangle OAC,
  cos  ∠AOC = 6/12 
  ÐAOC = 1.047 rad (change calculator to Rad mode)
  ÐBOC = 1.047 rad

(b) 
Area of the shaded region
= Area of BOC – Area of triangle AOC
= ½ (12)2 (1.047) – ½ (6) (12) sin 1.047 (change calculator to Rad mode)
= 75.38 – 31.17
= 44.21 cm2

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