Imagine a square surface ABCD touching a spherical surface with centre O at some point on the plate(not necessarily the centre of the plate). Join OA,OB,OC and OD. Thus, O-ABCD forms, in general, an oblique pyramid. Where should the point of contact lie for the volume of the part of pyramid that lies outside the spherical surface to be minimum? Intuitively, it must lie at the center of the plate(I think).

I am unknown to 3-D Geometry. Can I use " a triangle a vertex at the centre of a circle and the opposite side (with a given fixed length) being tangential to the circle" as a 2-D analogue of this problem and try to minimize the area of the triangle external to the circle(in which case, the point of contact must be the centre of the base of the triangle...I think).

This analogue if retraced into a 3-D problem would involve a triangular prism (and not a pyramid) and a cylinder(and not a sphere) with the cylinder touching the base surface of the prism in a line and one ege of the prism coinciding the axis of the cylinder(that axis which does not intersect the curved surface of the cylinder). Any analytical solution(not involving rigors of solving a differential equation)???

I hope i am clear in conveying my ideas. This wild fragment of imagination has just popped up in my head out of nowhere. But nevertheless coaxes me to think. plz help.