If it is AB//CD, the answer is

Let e be EF⊥BC, the vertical foot be f, the length of EF be l, and the top, bottom and height of trapezoid be a, b and h respectively.

Then a+b+h=AB+BC+CD=6.

L=(a+b)/2, that is 2L+h=6 (1).

Triangle BEF is RT triangle.

L 2+(h/2) 2 = be 2 = 5, that is, (4l) 2+h 2 = 20 (2).

It can be obtained from (1) 2-(2):

4Lh=6^2-20= 16

Lh=4

So:

Trapezoidal area s = (a+b) h/2 = LH = 4.