Maximum Area of Rectangle Inscribed in Semicircle

What is the maximum area of a rectangle that can be inscribed in a semicircle of radius  
Let the triangle OBC subtend an angle  
  at the centre of the circle.
The triangles ABO abd CDO have the same area by symmetry, and together subtend an angle of  
\[\pi - \theta\]
. We can consider them to form a triangle with sides  
  from the centre of the circle to the edge, and subtending an angle of  
\[\pi - \theta\]
  at the centre.
The area of the rectangle is then
\[\begin{equation} \begin{aligned} \frac{1}{2} r^2 sin \theta + \frac{1}{2} sin (\pi - \theta) &= \frac{1}{2} r^2 (sin \theta + sin (\pi - \theta )) \\ &= \frac{1}{2}r^2( sin \theta + sin \theta ) \\ &= r^2 sin \theta \end{aligned} \end{equation}\]

The maximum value of  
\[sin \theta\]
  is 1, when  
\[\theta = \frac{ \pi}{2}\]
  so the maximum area of the rectangle is  

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