Problem

In the diagram, $AOD$ and $BC$ are two parallel straight lines. Arc $AB$ is part of a circle with centre $O$ and radius $15$ cm. Angle $BOA = \theta$ radians. Arc $CD$ is part of a circle with centre $O$ and radius $10$ cm. Angle $COD = \frac12\pi$ radians.

(a) Show that $\theta = 0.7297$, correct to $4$ decimal places.

(b) Find the perimeter and the area of the shape $ABCD$. Give your answers correct to $3$ significant figures.