# Spherical trigonometry has no identified invariant

For plane trig we have Law of Sines

$$\frac{ a}{\sin A} =\frac{ b }{\sin B} =\frac{ c }{\sin C}= 2 R$$

where $2R$ is the diameter of the circum-circle.There is a clear understanding of the relationship of lengths and angles of a plane triangle.

For spherical trig also, we have a Law of Sines

$$\frac{ \sin a}{\sin A} =\frac{ \sin b }{\sin B} =\frac{ \sin c }{\sin C}=What?$$

where the invariant 3D object is geometrically unknown (at least to me). Just we know it is a number. There is no clear or complete understanding of the relationship of lengths and angles of spherical triangle.

May be it can be expressed in 4D geometry. Or, is it related to the $small$ circle's integral curvature circumscribing the vertices? I could not find reference in Todhunter/Leathem's (century old) text book.

$$\frac{\sin a}{\sin A}= \frac{\sin a \sin b \sin c} {\sqrt{1-\cos^2 a-\cos^2 b-\cos^2 c+2\cos a \cos b \cos c}}$$