Area from cross product (diagonal vectors)
- Paul Miller
What is the area for a region contained by four points? This describes a general quadrilateral. The answer is surprising: the area between four points is equal to the area of a related triangle given simply by the diagonals. If we move the diagonals parallel to themselves, the enclosed area stays the same (reasoning: one diagonal acts as a 'common base' and the second diagonal is 'split' into a height above and a height below. So the two triangles combine to give a fixed total area.) So, we can move either vector anyway - so long as it does not change direction - including moving them 'foot to foot' so that they form a single triangle rather than a quadrilateral. MOVE the indicated points around and see how the area (which GG recalculates) stays the same in the case of the triangle and the quadrilateral sharing the same vectors!