**Example 011**

A layer of equal spheres is in the form of a square. The spheres are arranged so that each sphere is tangent to every one adjacent to it. In the spaces between sets of 4 adjacent spheres, other spheres rest, equal in size to the original. These spheres form in turn a second layer on top of the first. Successive layers of this sort form a pyramidal pile with a single sphere resting on top. If the bottom layer contains 16 spheres, what is the height of the pile in terms of the common radius r of the spheres?

**Solution 011**

The lines that trace the center of the spheres will form into square pyramid. See figure below.

From the pyramid shown to the right:

$(2x)^2 = (6r)^2 + (6r)^2$

$(2x)^2 = 2(36r^2)$

$2x = 6r\sqrt{2}$

$x = 3r\sqrt{2}$

$x^2 = 9r^2(2)$

$x^2 = 18r^2$

$h^2 + x^2 = (6r)^2$

$h^2 + 18r^2 = 36r^2$

$h^2 = 18r^2$

$h^2 = (9r^2)(2)$

$h = 3r\sqrt{2}$

Total height of the pile

$H = 2r + h$

$H = 2r + 3r\sqrt{2}$

$H = (2 + 3\sqrt{2})r$ *answer*