We determine the value of a to be 0.268 radians which will give 1/3 the area of the circle.

Input := 


eq3 = cut[a]/R^2 == 1/3 Pi R^2/R^2

Output =


                  Pi

2 a + Sin[2 a] == --

                  3

Input := 


sol = FindRoot[eq3,{a,1}]

Output =


{a -> 0.268133}

Input := 


tsol = a/.sol[[1]]

Output =


0.268133