You need a rectilinear polygon that tessellates, and has the fewest sides possible to minimize the number of cuts necessary. And it would probably help the cutt

You need a rectilinear polygon that tessellates, and has the fewest sides possible to minimize the number of cuts necessary. And it would probably help the cutting if the shape is entirely convex, so that cuts can overshoot a bit without damaging anything. That suggests a rectangle is the only possible shape.