Signed 2D Triangle Area from the Cross Product of Edge Vectors
The signed area of a triangle in the - plane with vertices , , is given by half the component of the cross product of the edge vectors and .
The edge vectors are
The cross product of the edge vectors is .
The signed triangle area is half the component of the cross product: ,
which is equivalent to half the determinant of the matrix formed by the coordinates of when it is embedded in 3D space at : .
 J. O'Rourke, Computational Geometry in C (Cambridge Tracts in Theoretical Computer Science), 2nd ed., Cambridge: Cambridge University Press, 1998.