Dr. Sarah's Area of a triangle on a sphere and the sum of angles of a triangle

Adapted from materials by John Sullivan
Do this assignment before reading about hyperbolic geometry.

We will experiment with paths on spheres. A straight path on a sphere is a great circle, like the equator, or a longitude. A straight path on any surface is usually called a geodesic.

Materials necessary:
Ball, masking tape, see thru beach ball and markers.

Do this with masking tape on a regular ball first, and then go back and use markers on the see thru beach ball. Take detailed notes as you go along (you will turn these in). Some details will be easier to see on the see thru beach ball.

Area of Triangle on a sphere

Draw one complete great circle, and then draw a second at a certain angle (call it x) to the first. Do these two paths intersect anywhere else? What are all the angles at all the points where they meet?

Call the total surface area of the sphere A. (The particular number might depend on the size of the sphere, or what units we used to measure area, so its easiest to simply give it the name A.) Can you tell what the area of each of the four regions on your sphere is, in terms of the angle x you started with? Try some specific example and then try the general case. Just look at what fraction of the sphere is in which region to deduce the areas; we won't try at this point to give a rigorous definition of area.

(These regions are called lunes, since at any time the portion of the moon which is both visible and lit is such a lune.)

By drawing a third great circle, we can make a triangle with any two other angles y and z. (Starting with a triangle, what we want to do is to extend each of the three sides to be a full great circle.) How many regions are there on the sphere now? What can you say about opposite regions?

Fix your attention on the original triangle, and call its area T. Its angles, remember, are x, y, z. Opposite the angle x is a side, and across this side is another region on the sphere---call its area X. Similarly, label the areas of the other adjacent regions Y and Z. (Be careful in your work to distinguish the upper- and lower-case letters!)

What are T+X, T+Y and T+Z in terms of x, y and z? What is T+X+Y+Z? What is the area T of the triangle?

The sum of the angles of a triangle on a sphere

  • Without using your formula from above, carefully prove that the sum of the angles of a triangle is greater than or equal to 180 degrees.
  • Use your formula from above to prove that the sum of the angles of a triangle is strictly greater than 180 degrees.

    Be prepared to present your solutions to any of the above to the class.