I did a presentation in one of the classes I'm taking last week. (The class is Geometric Structures on Manifolds with Professor Adeboye, and it's a blast, both because of the instructor and the material!) I was supposed to give some sort of hour long overview of spherical geometry, which I almost never think about. My professor pointed me to several resources and I learned a lot, but the thing I was most excited about was the area formula for triangles on the sphere. I know the formula, but I had never actually derived it. I thought it was really cute and shared it with several people last week, so now I'm sharing it with you!
Say you have a triangle T on the unit sphere with angles as shown. The sides of the triangle are made up by line segments of three great circles (geodesics), which I've drawn in different colors.
A couple of observations:
The great circles form another triangle with the same angles and vertices antipodal from our original triangle. (Also shaded)
The triangle is the intersection of three 'wedges' of the sphere, one for each of the three angles. (And the same for it's antipodal triangle)
These wedges are much easier to figure out the area of, so we look at those instead! Take w(a) to be the double wedge associated with angle a of our given triangle. Each of the wedges is shown below.
More observations:
The union of the three double wedges is the whole two-sphere.
The intersection of the three double wedges is our triangle T and its antipodal copy - this means that the sum of the areas of the double wedges triples counts each triangle. That is,
Let's find the area of w(a). Rotating so that vertex is at the north pole, we have a picture like the one on the left. The ratio of w(a) to Area(S) should be the same as the ratio of 2a to the total angle. That is,
We know the area of the unit sphere! So we can then write the area as
This holds for any of the angle we chose, and so we also know that
Area(w(b)) = 4b and Area (w(c)) = 4c. Now, we can go back to the formula from earlier and plug these values in.
Now, we can look at the triangle from before with new eyes!
Takeaways:
Just like in hyperbolic space, we only need to know the angle sum of a triangle to find its area.
Also, triangles on the sphere are 'fat': The angle sum of a triangle must have angle sum greater than pi.
Note: This is the treatment given in the The Shape of Space by Jeffrey Weeks.