Question

Below fig. shows a sector of a circle, centre O. containing an angle θ°. Prove that

(i) Perimeter of shaded region is r(tan θ+secθ+(πθ/180)−1)

(ii) Area of shaded region is (r²/2)(tanθ−πθ/180)

Answer 1

Given angle subtended at centre of circle = θ

∠OAB = 90° [At joint of contact, tangent is perpendicular to radius]

OAB is right angle triangle

Perimeter of shaded region = AB + BC + (CA arc)

Area of shaded region = (area of triangle) – (area of sector)