Hello, Jenny!

A different approach . . . (and a different answer)

A bowling ball of radius is placed inside a box just large enough to hold it,

and it is secured for shipping by a Styroform sphere into each corner of the box.

Find the radius of the largest Styrofoam sphere that can be used. Code:

A R B
*-------*-*-*-------*
| * : * / |
| * : / * |
|* R: / *|R
| : / |
* : / *
* * - - - - *C
* O R *
| |
|* *|
| * * |
| * * |
*-------*-*-*-------*

In the upper-right, we have an R-by-R square:

The diagonal has length

Diagonal intersects the circle at (not shown).

Then: .

This is the diagonal of a smaller square in the upper-right corner.

. . Its side is: .

The radius of the inscribed circle is: .

Therefore: .