2/6/04                                            Triangles on a Sphere

1. On a sphere draw 3 great circles all perpendicular to one another.

• -3 line segments from pts of intersection of the triangles form a triangle.  We have 8 total  equilateral triangles formed.
• -The angle of each angle within the triangles are 90°  due to the fact that all the great circles we made are perpendicular and thus 90°.
• -Since each angle is 90° then the sum of all three angles is 90°+90°+90°=270°.  Hmmm very interesting the sum of the angles(270°)>180°, which we normally think of as a constant angle sum for a triangle in Euclidean space.

2.  Construction of an Equilateral Triangle (There are two simple ways to do this.)

• -Pick a point A and draw some non-great circle centered at A which we’ll call circle a.
• -Pick another point B which is on the circle a.
• -Construct a second circle centered at B with the same radius as circle a which we’ll call circle b

or

•  -Pick two points A and B and construct a circle with radius AB which we’ll call circle a

  -Note:  Make sure its not a great circle

• -Construct a second circle centered at B with radius AB which we’ll call circle b

-With either construction you can connect points A, B and one of the intersections of circles a and b. This should construct an equilateral triangle with sides equal to AB on the sphere.

3. In class we all used different sized circles and thus different lengths for AB.  Let's compile those measurements into the following table:

We can see that all the angle sums are greater than 180°.  We can also see that as the side length goes up so does the angles sum. As the lengths of the sides get smaller is the angle sum of triangle heading towards some limiting value?  Think about the surface the triangle would lie on when the side lengths were extremely small.  Could the plane be pretty close to a Euclidean plane?  What would the angle sum of the triangle be then?  Just something to think about.