Math 402, Assignment 9, Spring, 2008

Due Wed. April 9 (at the beginning of class)

1. Consider the formula giving the area of a triangle on the sphere in terms of interior angles. In class, we proved this formula for ``small triangles'', that is, triangles which were not linear so that every side was less than half the circumfrence of the sphere. For each of the cases below, decide if the same formula works. If it doesn't work, give a formula for the area of the triangle in terms of angles that does work, and explain your reasoning. (Remember that we have defined the interior angle of a triangle ABC at the vertex A to be the angle formed by the rays AB and AC. In particular, it is always at most 180 degrees and doesn't depend on which side we call the "interior".)
   
   1. Linear triangles
   2. Lunar triangles
   3. Witch's hat triangles
   
   
2. Consider a convex polygon on E^2, H^2, or S^2 with n sides. Find a formula for the sum of the interior angles of the polygon in terms of n and the area of the polygon. Prove your claim. Hint: use induction.

3. Make some cones with different cone angles. Make sure some are bigger than 360 degree angles. Draw some triangles so that the cone point is inside the triangle. What is the sum of the interior angles? Write down a formula which expresses the sum of the interior angles as a function of the cone angle.

4. Warning: For this last problem, you cannot use the angle sum formulas. Assume that there exists some constant C so that the sum of the interior angles of every triangle is C. Prove that C = 180 degrees. Hint: Divide the triangle in two.