Area Of Sector And Segment SS1 Mathematics Lesson Note

AREA OF SECTORS

The area of a sector of a circle is given by the formula;

Area of sector = θ    x  πr 2 ÷ 360°

where    r = radius of the circle, θ = angle subtended at the centre by XY or angle of the sector

Examples

Calculate the area of a sector of a circle which subtends an angle of 45° at the centre of the circle, diameter 28cm (π = 22/7).

The area of a circle PQR with centre O is 72cm2. What is the area of sector POQ, if POQ  = 40°? 

Solutions

1. Since the diameter of the circle = 28cm 

         d  = 2r  = 28

where d = diameter and r = radius

thus 2r = 28

 2r     =  28   = 14cm

2 

Area of sector =   θ    x    πr 2 ÷ 360°

=   45   x  22 x  ( 14 ) 2

     360       7

=  1/8  x 22/7  x 14 x 14  cm

         = 77cm2

2. Since the area of the whole circle  PQR = 72cm2 

Then 

Area of sector = θ    x    πr2 ÷ 360°

But πr2=  Area of the whole circle PQR = 72cm2

:. Area of          =   40     x 72cm2

sector POQ      ÷   360°

    = 8cm2

AREA OF SEGMENTS

A segment of a circle is the area bounded by a chord and an arc of the circle. Considering the figure below, we have a major segment and a minor segment.

In a diagram, the Area of the shaded segment= Area of sector POQ – Area of triangle POQ = θ

360°   x πr2  –  ½ r2 sin θ

Where 

r = radius of the circle

θ = angle subtended by the sector at the centre

Π=  a constant = 22/7 

ASSIGNMENT 

1. Calculate the  area of a sector of a circle of radius 6cm which subtends an angle of 70° at the centre (π = 22/7)      A. 44cm2        B. 22cm2        C. 66cm2   D. 11cm2   E. 16.5cm2
2. What is the angle subtended at the centre of a sector of a circle of radius 2cm if the area of the sector is 2.2 cm2? (π = 22/7)A. 120°    B. 31 ½°  C. 43° D. 58°  E. 63°
3. What is the radius of a sector of a circle which subtends 140° at its centre and has an area of 99 m2?     A  18m    B.   27m        C. 9m        E. 30m        E. 24m
4. A sector of 80° is removed from a circle of radius 12cm What area of the circle is left?     A. 253cm2   B. 704cm2C 176cm2D. 125cm2 E. 352cm2π