In the diagram, $AOB$ is a quarter circle with centre $O$ and radius $r$. The point $C$ lies on the arc $AB$ and the point $D$ lies on $OB$. The line $CD$ is parallel to $AO$ and angle $AOC=\theta$ radians. Express the perimeter of the shaded region in terms of $r, \theta$ and $\pi$.

In the diagram, $AOB$ is a quarter circle with centre $O$ and radius $r$. The point $C$ lies on the arc $AB$ and the point $D$ lies on $OB$. The line $CD$ is parallel to $AO$ and angle $AOC=\theta$ radians. For the case where $r=5 \mathrm{cm}$ and $\theta =0.6$, find the area of the shaded region.

In the diagram, $AB$ is an arc of a circle with centre $O$ and radius $4 \mathrm{cm}$. Angle $AOB$ is $\alpha$ radians. The point $D$ on $OB$ is such that $AD$ is perpendicular to $OB$. The arc $DC$, with centre $O$, meets $OA$ at $C$. Find an expression in terms of $\alpha$ for the perimeter of the shaded region $ABDC$.

In the diagram, $AB$ is an arc of a circle with centre $O$ and radius $4 \mathrm{cm}$. Angle $AOB$ is $\alpha$ radians. The point $D$ on $OB$ is such that $AD$ is perpendicular to $OB$. The arc $DC$, with centre $O$, meets $OA$ at $C$. For the case where $\alpha =\frac{1}{6}\pi$, find the area of the shaded region $ABDC$, giving your answer in the form $k\pi$, where $k$ is a constant to be determined.

The diagram shows a metal plate made by fixing together two pieces, $OABCD$ (shaded) and $OAED$ (unshaded). The piece $OABCD$ is a minor sector of a circle with centre $O$ and radius $2r$. The piece $OAED$ is a major sector of a circle with centre $O$ and radius $r$. Angle $AOD$ is $\alpha$ radians. Simplifying your answers where possible, find in terms of $\alpha , \pi$ and $r$, the perimeter of the metal plate.

The diagram shows a metal plate made by fixing together two pieces, $OABCD$ (shaded) and $OAED$ (unshaded). The piece $OABCD$ is a minor sector of a circle with centre $O$ and radius $2r$. The piece $OAED$ is a major sector of a circle with centre $O$ and radius $r$. Angle $AOD$ is $\alpha$ radians. Simplifying your answers where possible, find in terms of $\alpha , \pi$ and $r$, the area of the metal plate. It is now given that the shaded and unshaded pieces are equal in area.

The diagram shows a metal plate made by fixing together two pieces, $OABCD$ (shaded) and $OAED$ (unshaded). The piece $OABCD$ is a minor sector of a circle with centre $O$ and radius $2r$. The piece $OAED$ is a major sector of a circle with centre $O$ and radius $r$. Angle $AOD$ is $\alpha$ radians. Simplifying your answers where possible, find in terms of $\alpha , \pi$ and $r$. Find $\alpha$ in terms of $\pi$.