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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?

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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.

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  • $\begingroup$ Thanks for the answer. So how would I go about describing a simple example, such as a triangle being dragged by its vertex along the area of another triangle? $\endgroup$ Jun 1 '15 at 11:34
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    $\begingroup$ Saying "a triangle, dragged by it's vertex along the area of another triangle" seems like a find description. If you meant "How would I compute the area of the resulting object?," I would note that it's a polygon, find its vertices, and then use one of the many "area of a polygon" formulas to compute the area. $\endgroup$ Jun 1 '15 at 12:40

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