Area of regular polygon triangles

[David Epstein]

If a regular polygon has 4 or 8 sides it seems to be inscribed so that all points touch the square defined by the object’s height and width, which means that it should be fairly easy to deduce the polygon’s area.  
But with 3 sides, the triangle appears slightly smaller than the maximum size that could fit into that square.  Is there some math that would give me the area of that triangle from the height and width of the square (i.e., the official height and width of the polygon object)?
David Epstein