3. The diagram below shows a badge \(O D C\). The shape \(O A B\) is a sector of a circle centre \(O\) and radius \(r \mathrm {~cm}\). The shape \(O D C\) is a sector of a circle with the same centre \(O\). The length \(A D\) is 5 cm and angle \(A O B\) is \(\frac { \pi } { 5 }\) radians. The area of the shaded region, \(A B C D\), is \(\frac { 13 \pi } { 2 } \mathrm {~cm} ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{e4a4ea5b-7278-4735-925c-265a556ad679-06_684_531_571_769}

1. Determine the value of \(r\).
2. Calculate the perimeter of the shaded region.