And we check the top piece to see if it is one third the area of the circle, or 1/3 Pi r^2 = 1.0472 r^2. It is!

Input := 


TopThird = 

	Integrate[Sqrt[r^2 - x^2] - r Sin[tsol],

				{x,- r Cos[tsol],r Cos[tsol]}] 

Output =


           2                    2

-0.510931 r  + 0.255465 r Sqrt[r ] + 

                          2

   2        3.63968 Sqrt[r ]

  r  ArcTan[----------------]

                   r

Input := 


TopThird/.{Sqrt[r^2]->r}

Output =


        2

1.0472 r