Four small spheres with radius of $2$ are tangent with one another. If all $4$ small spheres tangent inside big sphere, find radius of the big sphere.

Question

Four small spheres with a radius of $2$ are tangent with one another. If all four small spheres tangent inside a big sphere, find the radius of the big sphere.

I can't solve it, but I think it involves advanced calculus.

Answer 1

The centers of the four spheres are the vertices of a regular tetrahedron of side length $a=4$. Its circumradius is given by

$ R = \dfrac{3}{4} h = \dfrac{3}{4} \sqrt{\dfrac{2}{3}} a = \sqrt{\dfrac{3}{8}} a = \sqrt{6}$

Therefore, the radius of the big sphere is $R + 2 = 2 + \sqrt{6}$