And now for the top area above the T. We see it too is 1/3 Pi R^2.

Input := 


TopArea = Integrate[Sqrt[R^2 - x^2] - R Cos[tsol],

			{x, - R Sin[tsol], R Sin[tsol]}]//N

Output =


           2                    2

-0.510931 R  + 0.255465 R Sqrt[R ] + 

                          2

   2        3.63968 Sqrt[R ]

  R  ArcTan[----------------]

                   R

Input := 


TopArea/.{Sqrt[R^2]->R}

Output =


        2

1.0472 R