ABSTRACT
Even though there is no overwhelming need for a proof of the formula you just discovered, I would like to include one anyhow because it is so simple and el egant. (It is not, however, the sort of thing you’re likely to stumble onto on your own. I struggled for hours without being able to prove the formula at all.)
First we have to know how to compute the area of a “double lune.” A double lune is a region on a sphere bounded by two great circles, as shown in Fig ure 9.3. The largest the angle a can ever be is tt , at which point the double lune fills up the entire sphere. So if a is, say, t t / 3 , then we reason that since t t / 3 is V3 the greatest possible angle tt , the double lune must
fill up V3 the area of the entire sphere, namely (V3)(4 7 r) = 4 77-/3 . Using the same reasoning, we get that the area of a double lune with angle a is (o J t t ) { A tt) = 4 a . You can check this formula for some special cases, e.g. a = 7t/2 or a = tt .
