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Question
In the given figure, an equilateral triangle has been inscribed in a circle of radius 4 cm. Find the area of the shaded region.
Solution
Draw
OD ⊥ BC
.
Because ΔABC is equilateral, ∠A = ∠B = ∠C = 60° .
Thus, we have:
∠OBD = 30°
`⇒ "OD"/"OB" = sin 30°`
`=> "OD"/"OB" = 1/2`
`=>"OB" = (1/2)`
`=> "OD" = (1/2xx4)"cm" [therefore "OB" = "radius"]`
⇒ OD =2 cm
⇒ BD2 = (OB2 - OD2) [By Pythagoras 'Therom']
⇒ BD2 = (42 - 22) cm2
⇒ BD2 = (16 - 4) cm2
⇒ BD2 = 12 cm2
`⇒ "BD" = 2sqrt(3)`
Also
BC = 2 × BD
`=>(2xx2sqrt(3))`
`=4sqrt(3)`
∴ Area of the shaded region = (Area of the circle) - (Area of Δ)
`=|(3.14xx4xx4)-(sqrt(3)/4xx4sqrt(3)xx4sqrt(3))|"cm"^2`
`= |50.24 - (12xx1.73)| "cm"^2`
= (50.24 - 20.76) cm2
= 29.48 cm2
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