**Known values**: Cartesian coordinates of all vertices (x_{1}, y_{1}), _{(}x_{2,} y_{2}), â€¦(x_{n}, y_{n}).

**Solution**: Area * A* of simple polygonal shape.

1. Subtract surrounding trapezoids (red) from encompassing trapezoids (green).

2. In general:

where

is the area of the polygon,**A**is the number of sides (or vertices) of the polygon,**n**are the Cartesian coordinates of polygonâ€™s vertices, and**x**_{i}, y_{i}, i=1, 2, â€¦, nor**+1***-1*define**next**orvertex from i vertex.*previous*

If vertices are numbered clockwise, the determinants of the expressions above are positive whereas if numbered counterclockwise, they are negative in which case absolute values should be considered.