Problem

The diagram shows a metal plate $ABCDEF$ consisting of five parts. The parts $BCD$ and $DEF$ are semicircles. The part $BAFO$ is a sector of a circle with centre $O$ and radius 20 cm, and $D$ lies on this circle. The parts $OBD$ and $ODF$ are triangles. Angles $BOD$ and $DOF$ are both $\theta$ radians.

(a) Given that $\theta = 1.2$, find the area of the metal plate. Give your answer correct to 3 significant figures.

(b) Given instead that the area of each semicircle is $50\pi\text{ cm}^2$, find the exact perimeter of the metal plate.