Usage patterns
〈物〉を平面上に配置するto arrange objects on a (flat) plane
Arrange the shapes on the plane and find their area.
〈物〉が平面と交わるfor something to intersect a plane
This line intersects the plane at a single point.
〈物〉を平面に投影するto project something onto a plane
Project the shape of the solid onto a plane to create a drawing.