How are determinants used to find areas of polygons?

Question

I need to know what this is

Answer 1

The polygon can be represented by points on a graph which are the vertices of the polygon. So for polygon with n sides we will have n vertices. We represent the coordinates of the vertices by (x_i,y_i) where 1≤i≤n.

Graphing the points enables us to see in which order we need to add n determinants. Starting at any vertex we move from vertex to vertex anticlockwise. This determines which coordinates are (x₁,y₁), (x₂,y₂), ..., (x_n,y_n). The area A of the polygon is given by half the sum of the determinants:

⎜x_i x_i+1⎟

⎜y_i y_i+1⎟

If we call this determinant D_i, then A=½(D₁+D₂+...+D_n-1).