Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated diameter with a normal coinciding with the bisector of $\theta$ when the planes are orthogonal (I think it is intuitively clear what this angle defines, even though the precise statement is a little long).
The lune on which the arc $AB$ is located has an area of $2r^2\theta_{\varphi=\pi/2}$, where $\theta_{\varphi=\pi/2}$ is the angle $\theta$ at $\varphi=\pi/2$. The angle $\theta$ is related to $AB$ by length$(AB) = r\theta$, but I would like to know the relation between $\theta$ and $\varphi$, once $\theta_{\varphi=\pi/2}$ has been fixed.
This is essentially so I can integrate some function from $\varphi=0$ to $\varphi=\pi$ to get the area of the lune. I need that to get an alternative definition, by integration, of the surface area of the $n$-sphere.
Edit: If it is not clear, the arcs from the north pole (topmost emphasized point) to $A$ and to $B$ are the same length, and $AB$ is a segment of a great circle (i.e. a geodesic).
Assuming that $\theta_{\varphi=\pi/2}$ is the angle between the great circle containing $N$ and $A$ and the great circle containing $N$ and $B$, and that $\varphi$ is the angle between the north pole and the great circle containing $A$ and $B$, then $\triangle N(\frac{A+B}{2})B$ is a right spherical triangle with the right angle at $\frac{A+B}{2}$. The basic relations for right spherical triangles yield $$ \tan\left(\frac{\theta_{\varphi=\pi/2}}{2}\right)\sin(\varphi)=\tan\left(\frac{\theta}{2}\right) $$ $\hspace{3.2cm}$