Hello, Jenny!

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

A bowling ball of radius $\displaystyle R$ 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 $\displaystyle r$ 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: $\displaystyle OABC.$

The diagonal $\displaystyle OB$ has length $\displaystyle R\sqrt{2}.$

Diagonal $\displaystyle OB$ intersects the circle at $\displaystyle D$ (not shown).

Then: .$\displaystyle DB \:=\:R\sqrt{2} - R$

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

. . Its side is: .$\displaystyle \frac{R\sqrt{2} - R}{\sqrt{2}}$

The radius of the inscribed circle is: .$\displaystyle \frac{1}{2} \times \frac{R\sqrt{2} - R}{\sqrt{2}} $

Therefore: .$\displaystyle r \;= \;R\left(\frac{\sqrt{2} - 1}{2\sqrt{2}}\right)$