Question

How are determinants used to find areas of polygons?

Answer 1

When you arrange the N points in sequence around the polygon (clockwise or counterclockwise), the area is half the magnitude of the sum of the determinants of the points taken pairwise. The N determinants will also include the one involving the last point and the first one.

For example, consider the vertices of a triangle: (1,1), (2,3), (3,-1). Its area can be computed as
  (1/2)*|(1*3-1*2) +(2*-1-3*3) +(3*1-(-1)*1)|
  = (1/2)*|1 -11 +4| = 3