One thing to do is to reduce the number of variables. So here is an extended hint. [I've assumed the base is on the diameter, but need not be the whole of it]
For example, there will be two points on the diameter - can you locate those, given that the area is maximum. Suppose you draw a random trapezoid. Try moving one of the points on the diameter and see whether it increases or decreases the area. Where does it need to be to get the biggest area? Do the same for the other point.
Then there are two points on a line parallel to the diameter. Where will these be if the area is a maximum?
Can you see that you need consider only one variable for the position of the line parallel to the diameter, and express your problem in terms of that?
Then you can use calculus on a one variable problem which you know how to tackle.