SOLVING SECTORS AND CIRCLES PROBLEMS

1. Look at the picture below! If the length of arc AB is equal to twice the length of radius r of the circle, the area of sector OAB in terms of the radius r will be equal to .....
   

   A. ½r2

   B. r2

   C. 2r2

   D. 4r2

   
   Solution :
   The area of sector AOB can be calculated by this formula :
   

   ⇒ Area AOB =	angle AOB	.Area of circle
   	360o

   ⇒ Area AOB =	angle AOB	.πr2
   	360o

   
   Now, we must find the angle AOB first.
   ⇒ Length of arc AB = radius x angle AOB
   ⇒ 2r = r. angle AOB
   ⇒ Angle AOB = 2 radians
   ⇒ Angle AOB = ²⁄_π.180^o
   
   Now, back to the formula above :
   

   ⇒ Area AOB =	2⁄π.180o	.πr2
   	360o

   ⇒ Area AOB =	2⁄π.1	.πr2
   	2

   ⇒ Area AOB =	1	.πr2
   	π

   ⇒ Area AOB = r²
   

   Answer : B

   
   
2. The angle of a sector in a given circle is 60 degrees and the area of the sector is equal to 18,84 cm². Calculate the arc length of the sector.

   A. 6,00 cm	D. 6,44 cm
   B. 6,02 cm	E. 6,82 cm
   C. 6,28 cm

   
   Solution :
   First, we can calculate the radius r by the formula below :
   

   ⇒ Area of sector =	60o	.πr2
   	360o

   ⇒ 18,84 =	1	.πr2
   	6

   ⇒ 113,04 = 3,14 r²
   ⇒  r² = 36
   ⇒ r = 6 cm.
   
   The arc length of the sector :
   ⇒ arc length of sector = radius x angle AOB
   ⇒ arc length of sector =  ⁶⁰⁄₁₈₀.π  x 6 cm 
   ⇒ arc length of sector = 6,28 cm.
   

   Answer : C

   
   
3. The angle of a sector in a given circle is 45 degrees and the radius is 8 cm. The area of sector is equal to .....

   A. 25,12 cm2	D. 26,22 cm2
   B. 25,42 cm2	E. 26,52 cm2
   C. 25,68 cm2

   
   Solution :
   

   ⇒ Area of sector =	angle of sector	.Area of circle
   	360o

   ⇒ Area of sector =	45o	.(3,14)(8)2
   	360o

   ⇒ Area of sector =	1	.(200,96)
   	8

   ⇒ Area of sector = 25,12 cm².
   

   Answer : A