I want to do a theoretical prediction before trying out an experiment. I just noticed that the arc separating the illuminated side of moon and the shadowed side is an ellipse arc. So Is it possible to write down the angle between Sun and moon (as seen from earth) in terms of the semi major and minor axes of that ellipse.
Yes. Assuming that the moon is distant enough to neglect parallax effects, the semi major axis is just the radius of the moon, $r$, and the semi minor axis is $|r\cos\theta|$ where $\theta$ is the Sun-Moon-Earth angle.
So when the sun is behind the Earth, the angle is 0 and the ellipse is a circle (full moon) so the semiminor axis = $r$. When the sun is at right angles to the Earth, the ellipse is a line, semiminor axis=0(half-moon) and when the sun is (nearly) behind the moon, the angle is 180, and again the semi-minor axis =$r$.