Example 2
Find the area of the segment AYB shown in figure, if radius of the circle is 21 cm and β AOB = 120Β°. (Use Ο = 22/7 ).
In a given circle,
Radius (r) = 21 cm
And, π½ = 120Β°
Now,
Area of segment AYB = Area of sector OAYB β Area of ΞOAB
Finding Area of sector OAYB
Area of sector OAYB = π/360Γ ππ2
= 120/360 Γ 22/7Γ(21)2
= 1/3Γ22/7 Γ 21 Γ 21
= 22 Γ 21
= 462 cm2
Finding area of Ξ AOB
We draw OM β₯ AB
β΄ β OMB = β OMA = 90Β°
And, by symmetry
M is the mid-point of AB
β΄ BM = AM = 1/2 AB
In right triangle Ξ OMA
sin O = (side opposite to angle O)/Hypotenuse
sin ππΒ° = ππ΄/π¨πΆ
β3/2=π΄π/21
β3/2 Γ 21 = AM
AM = βπ/π Γ 21
In right triangle Ξ OMA
cos O = (π πππ ππππππππ‘ π‘π πππππ π)/π»π¦πππ‘πππ’π π
cos ππΒ° = πΆπ΄/π¨πΆ
1/2=ππ/21
21/2 = OM
OM = ππ/π
From (1)
AM = π/πAB
2AM = AB
AB = 2AM
Putting value of AM
AB = 2 Γ β3/2 Γ 21
AB = β3 Γ 21
AB = 21βπ cm
Now,
Area of Ξ AOB = 1/2 Γ Base Γ Height
= π/π Γ AB Γ OM
= 1/2 Γ 21β3 Γ 21/2
= (πππβπ)/π cm2
Therefore,
Area of the segment AYB = Area of sector β Area of β π΄ππ΅
= (462 β πππ/π βπ ) cm2

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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