an equilateral triangle abc of side 6 cm has been inscribed in a circle . find the area of the shaded region . thai is the remaining part of the circle

Question

an equilateral triangle abc of side 6 cm has been inscribed in a
circle find the area of the shaded region thai is the remaining
part of the circle

Santosh Kumar · Accepted Answer

Draw a perpendicular bisector OP from point O to BC.

AB = BC = CA = 6 cm

Area of ΔABC = $\frac{\sqrt{3}}{4} (6)^{2} = 9 \sqrt{3} {cm}^{2}$

∠BAC = 60° (equilateral triangle)

BP = PC = $\frac{BC}{2} = \frac{6}{2} = 3 cm$

In ΔBOC,

∠BOC + ∠OBC + ∠OCB = 180°

⇒ 120° + ∠OBC + ∠OBC = 180°

⇒∠OBC = $\frac{180 ° - 120 °}{2} (∵ OB = OC)$

⇒∠OBC = 30°

$\cos 30 ° = \frac{BP}{OB} \Rightarrow \frac{\sqrt{3}}{2} = \frac{3}{OB} \Rightarrow OB= \frac{6}{\sqrt{3}} cm$

∴ Radius of the circle = $\frac{6}{\sqrt{3}} cm$

area of the circle $= π× (\frac{6}{\sqrt{3}})^{2}$

$= \frac{π× 36}{3} = 12 π {cm}^{2}$

∴ Area of the remaining (shaded) part = Area of the circle – Area of the equilateral triangle

$12 π - 3 \sqrt{3} = 37.68 - 15.57 = 22.11 {cm}^{2}$