1.56: Sector Area and Perimeter

Figure 1 shows a sector $\displaystyle OAB$ of a circle where $\displaystyle \angle AOB=\theta$ radians.

The circle has centre $\displaystyle O$ and radius $\displaystyle 15\,\mathrm{cm}$.

The point $\displaystyle C$ divides $\displaystyle OA$ in the ratio $\displaystyle 2:1$ and the point $\displaystyle D$ divides $\displaystyle OB$ in the ratio $\displaystyle 2:1$.

The area of the region $\displaystyle ABDC$, shown shaded in Figure 1, is $\displaystyle 100\,\mathrm{cm}^2$.

Find

(a) the value of $\displaystyle \theta$,

(b) the perimeter, in $\displaystyle \mathrm{cm}$, of the region $\displaystyle ABDC$.