In the adjoining figure O is the centre of the circle with radius r. AB, CD and EF are the diameters of the circle. ∠OAF = ∠OCB = 60°. What is the area of the shaded region?
Answers
Given : O is the centre of the circle with radius r. AB, CD and EF are the diameters of the circle. ∠OAF = ∠OCB = 60°.
To Find : What is the area of the shaded region?
Solution:
∠OAF = 60°
OA = OF = Radius
=> ΔOAF is Equilateral Triangle
∠OCB = 60°
OC = OB Radius
Hence ΔOCB is Equilateral Triangle
∠AOF = 60° , ∠BOC = 60°
=> ∠COF = 180° - 60° - 60° = 60° as AC is straight Line
∠DOE = ∠COF ( vertically opposite angle )
∠DOE = 60°
ΔODE is also an equilateral Triangle
Each sector has 60 ° angle
Area of shaded region = (60/360)πr² - (√3/4) r²
= r² (π/6 - √3/4)
= (r²/6) (π - 3√3/2)
Area of 3 shaded regions
= 3 (r²/6) (π - 3√3/2)
= (r²/2) (π - 3√3/2)
(r²/2) (π - 3√3/2) is the correct answer
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