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.

Answer 1

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