Calculating the Area of a Country Using Green's Theorem

Green's theorem shows the relationship between the length of a closed path and the area it encloses.

From a central locator point , 250 vectors run toward points on the country's border. The total area of the 250 triangles defined by pairs of successive vectors is equal to the enclosed country's area.

Using determinants, the area of the triangle is equal to . The total of the enclosed area is .

If the signed area of a triangle (determinant) is positive, the triangles are colored blue; if it is negative, they are red.

When the number of triangles is 250 and the perimeter is completely closed, it can be verified that the total area (i.e., the sum of the signed areas) of the triangles is independent of the locator position.

Snapshot 1: Shows the total area being independent of the position of the locator.

Snapshot 2: Shows the mostly positive areas (determinants) if the locator is in the middle. The direction of Mathematica's CountryData coordinates is clockwise.

Snapshot 3: This illustrates the red positive areas and the blue negative ones. An intermediate color is generated when they overlap.