The shaded area is a circle segment, which is obtained by subtracting the area of the triangle AOB from sector AOB.

Area of sector AOB
= ½r²θ (radians)
or
πr² x θ/360° (degrees)

where θ is the angle AOB, and also the angle of sector AOB

Area of triangle AOB = ½ ab sin c
= ½(OA)(OB)sinθ
= ½r² sinθ

(since the angle AOB is θ and OA and OB are radii of the circle so they are equal to r)

So area of segment
= ½r²θ - ½r²sinθ (radians)
= ½r²(θ - sin θ)

Or

πr² x θ/360° - ½r²sinθ (degrees)


Now you are given the arc length (la)

Arc length = (angle of sector)/360° x circumference of circle

= θ/360° x 2πr (degrees)

Or θ/2π x 2πr
= rθ (radians)


Using radians :

Since you are given la,

la = rθ

So θ = la/r

Area of segment
= ½r²(θ - sin θ) (from above)

= ½r²(la/r - sin(la/r) )

= ½rla - ½r² sin (la/r)


Using degrees :

la = θ/360° x 2πr
So θ = la/2πr x 360°

And θ/360° = la/2πr


Area of segment

πr² x θ/360° - ½r²sinθ

= πr² (la/2πr) - ½r² sin(la/2πr x 360°)

= ½rla - ½r² sin (180°la/πr)

hi, thank you! Actually Idk how to do a(ii)

Draw a line perpendicular to AB from O.
Call the foot of the perpendicular C.

Notice that now you have divided the isosceles triangle OAB into two congruent right angled triangles, AOC and BOC.

Now the perpendicular is also the angle bisector of angle AOB (we call this θ as before).

So angle AOC = angle BOC = ½θ

It is also the perpendicular bisector of AB so AC = BC = ½l

Also recall that OA and OB are radii of the circle (r)

Using trigonometry,

sin ∠AOC = opposite/hypotenuse = AC/OA

sin ½θ = ½l/r = l/2r

½θ = sin-¹ (l/2r)

Then θ = 2sin-¹ (l/2r)

Now area of segment as before

= ½r²(θ - sin θ)

= ½r² ( (2sin-¹ (l/2r) - sin (2sin-¹(l/2r)) )

= r² sin-¹ (l/2r) - ½r² sin(2sin-¹(l/2r))