# a sector of OAP of a circle with centre O, containing angle theta. AB is perpendicular to the radius OA and meets OP produced at B. prove that the perimeter of shaded region is r[tan theta + sec theta + ¶theta/180 -1]

AB = OA tan thetha = r tan thetha

cos thetha = r/OB

OB = r/ cos thetha = r sec thetha

therefore BP = r sec thetha - r

PA = pie. r. thetha/180

therefore required perimeter = r tan thetha + pie. r. thetha/180 + r sec thetha -r = r( r tan thetha = pie. r. thetha/180 = sec thetha - 1)

proved.