a) $AS = r\tan \theta $

Area $OAB = {r^2}\tan \theta $ or $\left( {OAS} \right) = {\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}{r^2}\tan \theta $

Area of sector $ = \frac{1}{2}{r^2} \times 2\theta \left( { = {r^2}\theta } \right)$

Shaded area $ = {r^2}\left( {\tan \theta  - \theta } \right)$ OE

b) $\cos \frac{\pi }{3} = \frac{6}{{OA}} \Rightarrow OA = 12$

$AP = 6$

$AS = 6\tan \frac{\pi }{3}\left( { \Rightarrow AB = 12\sqrt 3 } \right)$

$Arc{\text{ }}\left( {PST} \right) = 12\frac{\pi }{3}$

Perimeter $ = 12 + 12\sqrt 3  + 4\pi $