Example 1. Place four equal small balls on a plane so that the line connecting their centers forms a square, with each side having two balls touching, and place one more equal ball on top of these four balls so that it touches all of them. Given the highest point of the top ball is a. Find the radius $x$ of the small balls.
Analyzing with the five ball centers as vertices, forming a regular tetrahedron $A-B C D E$ with edge lengths of $2 x$ (this figure), the height $A O$ $=\sqrt{2} x$. Extending upwards and downwards by the radius $x$ of the small balls, the distance from the highest point $P$ of the upper small ball to the ground is then
$$
P Q=2 x+\sqrt{2} x=a \text {. }
$$

Solving for $x$ yields $x=\left(1-\frac{\sqrt{2}}{2}\right) a$.