Thus the area of middle section ABCD is two times the sum of the areas of the four congruent triangles BOP, EOB, AOE, AOQ and the two pieces BPM and AQN.


In our analysis we use t for the angle q (theta)

Input := 


BOP = 1/2 r Cos[t] r Sin[t];

Input := 


BPM = Integrate[Sqrt[r^2 - x^2],{x,r Cos[t],r}]

Output =


    2

Pi r

----- - (r (r ArcTan[

  4

                              2       2

          Cot[t] Csc[t] Sqrt[r  Sin[t] ]

          ------------------------------] + 

                        r

                    2       2

       Cos[t] Sqrt[r  Sin[t] ])) / 2

Input := 


MidPiece = 2 (4 BOP + 2 BPM) 

Output =


      2

2 (2 r  Cos[t] Sin[t] + 

           2

       Pi r

    2 (----- - (r (r ArcTan[

         4

                                   2       2

               Cot[t] Csc[t] Sqrt[r  Sin[t] ]

               ------------------------------] + 

                             r

                         2       2

            Cos[t] Sqrt[r  Sin[t] ])) / 2))