The diagram above shows four identical circles, each having a flat radius $r$ (i.e. flat area $\pi r^2$), touching one another at six different points (i.e. each of four identical circles touches rest three at three different points) on a spherical surface with a radius $R$. How to find out the radius $R$ (of sphere) in terms of radius $r$ (of circles)? Any help is greatly appreciated.
On
The six points where a pair of circles touch are the vertices of a regular octahedron, with each face inscribed in one of the circles. Picking these vertices to be the standard basis vectors and their negatives, it's easy to express the radius $R$ of the sphere as a multiple of the radius $r$ of the circles.
The six points of intersection of any couple of circles give a octahedron with side length $\sqrt{3}\, r$. Since the octahedron is inscribed in the sphere, $$\sqrt{3}\,r = \sqrt{2}\,R$$ follows from the pythagorean theorem.