The original tetrahedron is regarded as a triangular pyramid (one of which is determined as the bottom surface), and the four balls are regarded as three bottom balls and a top ball.
The center of the upper sphere is a parallel plane at the bottom of the triangular pyramid. The truncated triangular pyramid is divided into upper and lower parts, and the upper part is divided into smaller regular tetrahedron.
Wherein that cent of the upper sphere is the center of a triangle with the small regular tetrahedron base.
The distance from the center of the upper sphere to one side of the smaller tetrahedron is 1 (actually, it is the distance from the center of the sphere to the tangent point).
Therefore, the height of the smaller tetrahedron is 3. Therefore, the height of the original tetrahedron is 4+(2/3) times that of the root number 3.