
When the coordinate of three points which are vertices of a triangle and known, the area of the enclosed region may be detremined. For the case shown, the area of the triangular region P1P2P3 can be obtained by adding the measures of the trapezoidal regions M1M3P3P1 and M3M2P2P3 and then subracting the measure of the trapezoidal region M1M2P2P1. Now, since the area of the interior of a trapezoidis equal to one-half the sum of the measures of the parallel sides times the measure of the altitude,
The final expression in parentheses is precicely the expantion of the determinant
Therefore, at least for the case shown in the figure, the area of the interior of the triangle is equal to one-half the value of this determinant. For other positions of the vertices relative to the coordinate axes the proof may differ somewhat in the details, but the result is the same.