In Fig. 9, is sho

The length of the arc AP is given as:

Now, ΔOAB is a right angle triangle (∵ the tangent is perpendicular to the radius through the point of contact)

Now,

⇒ AB = OA× tanθ

⇒ AB = r× tanθ

Also, in right Δ OAB,

⇒ OB = OA × secθ

⇒ OB = r× secθ

Now, PB = OB-OP

⇒ PB = r.secθ – r

∴ The perimeter of the shaded area = Arc AP + AB + PB

= + r. tanθ + r.secθ – r

=