# Area of a 2D shape dragged along a 2D contour

Imagine you have a two-dimensional path (e.g., a circle) and you 'drag' another shape, say a triangle, along this contour. This creates a ring, which has an area.

My question: what is the general means of measuring the area of a shape dragged along a path? And furthermore, what if the shape is dragged across each point of a geometric object (say, a triangle vertex placed at each point in another rectangle), what would the resulting area be?

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.