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The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 8 cm . Angle \(A O B = \theta\) radians.
- In the case where \(\theta = 1\), calculate the area of the sector BOC.
- Find the value of \(\theta\) for which the perimeter of sector \(A O B\) is one half of the perimeter of sector BOC.
- In the case where \(\theta = \frac { 1 } { 3 } \pi\), show that the exact length of the perimeter of triangle \(A B C\) is \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).