The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of " p ", " q " and " r ".
42.
Hyperbolic trigonometry as usually studied uses the hyperbolic functions cosh, sinh etc ., and this contrasts with spherical trigonometry which uses the Euclidean trigonometric functions cos, sin, but with spherical triangle identities instead of ordinary plane triangle identities.
43.
While the measures of the internal angles in planar triangles always sum to 180? a hyperbolic triangle has measures of angles that sum to less than 180? and a spherical triangle has measures of angles that sum to more than 180?
44.
The area of a spherical triangle is proportional to its excess, by Girard's theorem the amount by which its interior angles add up to more than 180? which is equal to the amount by which its exterior angles add up to less than 360?
45.
Any of these points on the Celestial Sphere can be joined by an imaginary line to the ( theoretical ) centre of the earth, and the point at which they meet the surface of the earth gives you a spherical triangle with the vertices P, Z and X.
46.
Hi all-my question is pretty much as in the title : why is it that, assuming we take the shorter of the 2 arc lengths between any 2 points on S ^ 2, the perimeter of any given spherical triangle is strictly bounded above by 2 \ pi?
47.
The book introduces the concept of spherical triangle ( figures formed of three great circle arcs, which he named " trilaterals " ) and proves Menelaus'theorem on collinearity of points on the edges of a triangle ( which may have been previously known ) and its analog for spherical triangles.
48.
The book introduces the concept of spherical triangle ( figures formed of three great circle arcs, which he named " trilaterals " ) and proves Menelaus'theorem on collinearity of points on the edges of a triangle ( which may have been previously known ) and its analog for spherical triangles.
49.
The congruence theorems side-angle-side ( SAS ) and side-side-side ( SSS ) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle ( AAA ) sequence, they are congruent ( unlike for plane triangles ).
50.
For example consider an equilateral spherical triangle with sides of 60 km on a spherical Earth of radius 6371 km; the side corresponds to an angular distance of 60 / 6371 = . 0094, or approximately 10 " 2 radians ( subtending an angle of 0.57?at the centre ).
