Radian Measure

 

 

 

Contents

 

Radian & Degree

 

Arc Length

 

Area of Sector

  

Quiz

 

 

Geometry Main Page 

Area of Sector

 

The area of a sector is given by A = ½r2q, where q is in radian and 'A' is the area.

Proof:
 
By proportion:
 
area of sector = angle in sector
area of circle      angle in circle
 
 
As s = r  , 
A= ½r2 q = ½rs 

Example 1
 
A sector cut from a circle of radius 5cm has a perimeter of 18cm.
Find the area of the sector.
Perimeter = 18cm
    5 + 5 + s = 18 cm
                s = 8cm
Area of sector = ½rs
                    = ½ x 5 x 8
                    = 20cm2

Example 2
 
In the diagram, arcs AB and CD are arcs of concentric circles, centre O. If OA is 8cm, AC is 4cm and the area of sector AOB is 16cm2, calculate 
a) ÐAOB in radian
b) the area and perimeter of the shaded region.
a) Let ÐAOB = q radian
Area sector AOB = 16 cm2
          ½(OA)2 q = 16
                     q = ½
 
b) Area sector COD = ½(OC)2 q
                           = ½ x 122 x ½
                           = 36cm2 
 
Area shaded region = 36cm2 - 16cm2
                            = 20cm2 
 
Perimeter = 4cm + 3cm + 6cm + 3cm
              = 16cm