This geometric Demonstration establishes that the area of a parallelogram bounded by vectors and is .
Use the sliders to see how various parallelograms can be transformed into ones of equal area with their bases on the axis. If the axis does not intersect the parallelogram, slide the triangular portion farthest from the axis toward it. If the axis does intersect the parallelogram, slide the triangular portion cut off by the axis to the farther end of the parallelogram.
In the examples shown, the height of the resulting parallelogram is and its base is determined by the axis intersection of the line joining to , which is easily seen to be . Thus the area is .
Clearly the same result holds if, due to the choice of triangular portion, the roles of (, ) and (, ) are interchanged.