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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