A spherical triangle is a shape created on a sphere's surface by three great circle arcs meeting at each of its three vertices pairwise. A few important facts regarding spherical triangles are:
-The sides are not lines; rather, they are the arcs of huge circles. -Every side is quantified by its arc length, which is commonly represented as an angle at the sphere's centre. -A spherical triangle's total angles are always larger than 180° and less than 540°. -Any two sides added together always equal more than the third side.
This the first time I have encountered spherical triangles. I'm fine with flat ones but rather think the mathematics for these, would be well above my level.
Presumably the angles are those between the tangents of the arcs at the vertices?
I wonder the significance of 540º. It is 9 X 60 but I can't help thinking there is more to it that that.
How would you calculate the area? I imagine the ordinary [Base X Height / 2] would fail here because it is a curved surface, so would it need some very advanced three-dimensional Integration?
Isn't the fourth property (Sum of 2 sides > than the third side) true for all trinagles, though?
The "angle" meant in your third item is the angle measured on the surface of the sphere (and "> 180°" means the surface has positive curvature/isn't flat), but you can certainly have a spherical triangle where each side subtends only a fraction of a degree (the "angle" you first mention), so that the sum of the angles describing the sides is well under 180°, or even 1°.
