Why is the area of parallelogram not equal to the product of the distances between the two pairs of parallel sides? A parallelogram is just a rectangle that has a triangle cut off from one side and added to the other; so then why is the area not equal to the product of the distances between the two pairs of parallel sides?
As you suggested, let's take a rectangle and cut a triangle from one side and add it to the other. Imagine the resulting parallelogram has two horizontal sides, and that the other two sides are "non-vertical" (else we'd still have a rectangle.) The distance between the two "non-vertical" sides would be measured perpendicularly to those sides; however the distance needed for the area would have to be measured horizontally, because that would have been the distance between those two sides when the shape was still a rectangle.