From right triangle BFC

From right triangle CEO

Area of trapezoid BCEG

Area of trapezoid ABCD

For maximum area, dA/dθ = 0:

From right triangle BFC

          answer
 

A solution to the same problem has been done by Geometry without the use of Calculus. In that solution, the location of the tangent point H has been pointed out by direct analogy. See how it was done by following this link: Solution by Geometry alone