Find the area of the minor segment of a circle of radius 42cm, if the length of the corresponding arc is 44 cm.

Given,

Radius of circle, r = 42 cm

Let the angle of sector be θ with corresponding arc AB

We know that length of an arc

Where, θ is the angle of sector to the corresponding arc and 'r' is radius of circle.

As the length of arc is 44 cm. Putting values we get,

⇒ θ = 60°

Also, In ΔOAB

OA = OB [radii of same circle]

∠OBA = ∠OAB [Angles opposite to equal sides are equal]

Also,

∠OBA + ∠OAB + ∠AOB = 180° [Angle sum property]

⇒ ∠OBA + ∠OBA + θ = 180°

⇒ 2∠OBA + 60° = 180°

⇒ ∠OBA = 60°

⇒ ∠OBA = ∠OAB = ∠AOB = 60°

⇒ OAB is an equilateral triangle.

And area of equilateral triangle with side 'a'

Also, area of a sector

Where, θ is angle of sector and 'r' is the radius of circle.

Area of sector OAB

Area of required segment = area of sector OAB - area of ΔOAB

= (924 - 441√3) cm²