Beach Ball Investigation for Non-Euclidean Geometry Topic Index | Geometry Index | Regents Exam Prep Center

Materials:  beach balls (or other larger balls)                  (could be done with balloons with some care)                  string, protractors, rulers or yardsticks Note:  The ball can be marked by the teacher with a permanent marker before the activity begins.  This will allow the ball to be used with several different groups/classes of students.  Mark the poles.  Mark two other points on the ball allowing for adequate distance between the points.  Mark the vertices of several different sized triangles on the ball and

draw the triangles using great circles to form the sides.   Label all of your points for easy reference for students' answers.

The shortest distance between two points on a sphere is along the arc of the great circle joining the points.  The shortest distance between points on any surface is called a geodesic.  In a plane, a straight line is a geodesic.  On a sphere, a great circle is a geodesic.

Student Tasks:

1.  Using the string, determine the length of the great circle of the spherical ball.  Pull the string tight to the ball between the two poles, to approximate a geodesic.  Record this length.

2.  Find the distance between the two designated, but non-connected, points on the ball.  (These points will not be the poles.)  Record this length.  Is the geodesic you used for this length unique, or are other geodesics possible for this measurement?

3.  Locate the vertices of each of the triangles on the ball.  Using a great circle as a geodesic, find  the lengths of the sides of the triangles.  Record the lengths for all of the triangles.

4.  To the best of your ability, use the protractor to measure the angles in each of the triangles.  Record the measurements for each triangle.

5.  Make a concluding statement about the relationship between the angels of a triangle on a sphere.

6.  A discussion of Euclidean geometry versus non-Euclidean geometry would follow.