The thing you're constructing is called a "Minkowski Sum", and there's no simple formula for the area generated. It's also easy to make mistakes in describing such sums. For instance, a triangle dragged around a circle generates something that's more or less ring-shaped (or "annular"), but it's not a proper annulus: the "thickness" of the ring generally varies from point to point; that's easiest to see if you take the triangle to be long and skinny.
To see that there can't be a simple formula, consider a line segment dragged $A$ along two equal-length line segments $B$ and $B'$, where $B$ is parallel to $A$ and $B'$ is perpendicular to $A$. In the first case, you get a longer line; in the second, you get a rectangle. The first has no area, the second has positive area. So any answer depends not only on the two objects, but on some kind of geometric relation between them as well.