I can tell you one way of doing this, but there may bean easier way which I do not see at the moment.
Draw a figure with the shorter base on top.
You will have 4 right triangles, 2 of which are isosceles right triangles, namely a larger one on the bottom and a smaller one on top. Remember that in an isosceles right triangle the ratio of the hypotenuse to the leg is √2
Let x be the length of the shorter base; the longer base is 2a - x.
The bottom triangle has legs = (2a-x)/√2 and altitude (2a - x)/2
The top triangle has legs x/√2 and altitude x/2
The altitude of the trapezoid is [(2a-x)/2] + x/2 = a
The area of the trapezoid is (1/2) * 2a * a = a2
