Spherical triangle area

Group 1

We already established in class that the area of a lune is (a/360)S where s is the small angle between the great circles making up the lune, and S is the surface are of the sphere. We can now use this information to find the area of a spherical triangle. Suppose we have a spherical triangle with interior angles a, b, and c. We can find the area of each lune that contains the triangle using our formula for lune area. It turns out that the total area of all three such lunes together is half the total surface are of the sphere, because we could find identical pieces on the opposite side of the sphere that together would fit like a puzzle to form a complete sphere. Unfortunately, if we just add the area of these three lunes together, we have added in the triangle surfaces are 3 times! Therefore, half the spheres area is really the sum of the 3 lunes minus twice the area of the triangle (to remove the overlap). Thus, we now have an equation:

(a/360)S + (b/360)S + (c/360)S – 2x = S/2

where x is the triangle’s area.

Now, we simply solve for x. . .

(a + b + c)(S/360) – 2x = S/2

2x = (a + b + c)(S/360) – S/2
2x = (S/2) ((a + b + c)/180 – 1)

x = (S/4)(((a + b + c)/180) – 1)

x = (S/4)((a + b + c – 180)/180)

x = S * (a + b + c – 180)/720