6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6c32000f-574f-473c-bd04-9cfe2c1bd715-12_487_784_292_644}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
In the design
- \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
- sector \(O F E\) is congruent to sector \(O A B\)
- \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
- \(A O F\) is a straight line
Given that the size of angle \(C O D\) is \(\theta\) radians,
- write down, in terms of \(\theta\), the size of angle \(A O B\)
- Show that the area of the logo is
$$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
- Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).