Sums of Angles and Areas of Triangles



Section 0: Starting NonEuclid

In order to run NonEuclid, you should go to one of the computer labs on campus (or from home) and use a Windows-based machine, equipped with a relatively modern version of Netscape (4.6 or higher) or Internet Explorer (version 5). While you can run the software from any computer equipped with one of these browsers, it sometimes doesn't work on other types of machines. If you are on a Macintosh or a Unix machine, you may have to fiddle a bit to get things to work properly (changing the size of the screen seems to fix problems). Start a browser and go to the address:

math.rice.edu/~joel/NonEuclid

If the browser that you are using is Java enabled, you will see a banner for NonEuclid 1999.8b. Click on this and the program will begin. Click okay in the information box that appears. Click on the View menu, pull down to Hyperbolic Model, and then select Upper Half-Plane. You may wish to resize the screen at this point.

Section 1: Triangles in Hyperbolic Space

Draw a triangle on the screen. If you have forgotten how to do this, refer back to lab 3, which you can get at:

http://www.math.uiuc.edu/~jms/m302/labs/labs/lab3.html

Measure the triangle as you did in lab 3.

Question 1: Is the sum of the interior angles of the triangle always 180 degrees? Always more? Always less? Move the triangle several times, noting what happens to the sum of the angles.

Try to construct triangles whose sum of angles is as small as possible and triangles whose sum of angles is as large as possible (drawing one triangle and just moving it around is a painless way to do construct lots of triangles). Pay attention to what is happenning to the area.

Question 2: What is the exact relationship between the sum of the interior angles of a triangle and its area? Fill in the blank at the end of this statement: If the sum of the interior angles of a triangle is S, then the area of the triangle is:_______________.