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]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+