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 = r θ 5.2 = 13 ( C O B ) C O B = 0.4  radian

(b)
cos C O B = O A O C cos 0.4 = O A 13  (change calculator to Rad mode) O A = 11.97  cm A B = 13 11.97 = 1.03  cm C A = 13 2 11.97 2 C A = 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  A O B = 1 2 r 2 θ = 1 2 ( 5 ) 2 ( 6 5 ) = 15 c m 2

(b)
Area of shaded region = 1 2 r 2 ( θ sin θ )   ( 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

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