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The Area of Triangles in Hyperbolic Geometry

One of the most surprising facts in hyperbolic geometry is that there is an upper limit to the possible area a triangle can have, even though there is not an upper limit to the lengths of the sides of the triangle.

In hyperbolic geometry, the sum of the angles of a triangle is always less than 180 degrees (PI radians). The amount less than 180 is called the defect. In 1794 (when only 17) Gauss discovered the following formula for the area of a triangle in hyperbolic geometry:

Thus the area is proportional to the defect, with the above proportionality constant (k is 1 for the model of the hyperbolic plane we're using). Now it's easy to see why there is an upper limit to the area of all triangles; namely, the defect measures how much the angle sum is less than 180. Since the angle sum can never get below 0, the defect can never get above 180. Therefore, the area of a triangle in hyperbolic geometry is:

Where alpha, beta, and gamma are the interior angles of the triangle.

The Applet

Below is an applet that lets you draw a triangle in the hyperbolic plane. The hyperbolic plane is embedded inside of a disk; the edge of the disk represents infinity. Click on three points in the disk to form a triangle, and drag them around to see how its area changes.

Notice:

As you drag a vertex to infinity (the boundary of the disk), the angle at that vertex goes to zero.
Since the boundary of the disk represents infinity, the sides of the triangle become infinitely long as the vertices get closer to it.

Acknowledgements

Applet by: Adam S. Rosien

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Created: Apr 04 1996 --- Last modified: Wed Nov 5 09:56:03 1997