Solutions received gave part of an insight into the problem but ommitted to consider triangles on the dotty grid that do not have a base of whole unit length. It is possible to draw triangles whose bases and heights are neither horizontal or vertical. These triangles have bases that are not a whole number or units. A complete solution needs to consider these.

Here is a synopsis of the solutions offered for the cases considered so far (i.e. it does not consider triangles that have non-horizontal bases):

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 infimite 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

It is possible to demonstrate that it is also possible to obtain triangles with multiples of half a sq unit but there is still work to do.