The smallest triangle it is possibkle to draw has a base of 1 unit and a height of 1 unit. So the smallest area is $ \frac{1}{2} $ sq. unit.

There are an infinite number of triangles that can be drawn with these diagonals (see the problem "Shear Magic" )There are two ways of creating a triangle of area 1 sq and with a horizontal base:

Base 1 unit; height 2 units

or

Base 2 units and height 1 unit, again

For an area of 2 sq units there are three families of triangles with a hoirizontal base::

Base 1 unit and height 4 units

or

Base 2 units and height 2 units

or

Base 4 units and height 1 unit

For each family there are an infinite number of triangles