Finding the area of a simple polygon

Let a simple polygon (without self-intersections, but not necessarily convex) be given, specified by the coordinates of its vertices in traversal order clockwise or counterclockwise. It is required to find its area.

Method 1

You can iterate over all edges and sum the areas of the trapezoids bounded by each edge. The area must be taken with a sign (it is precisely thanks to the sign that all the “extra” area will cancel out).

Formula for one edge $(X_1, Y_1) \rightarrow (X_2, Y_2)$ :

Finding the area of a simple polygon

Method 1

Method 2