Question 727397
The diameter of the ball is the same as the width of the box, and 2 times the radius.
The formula for volume of a sphere says that a ball of  the radius 
{{{r}}} has a volume of
{{{V(ball)=(4/3)*pi*r^3}}}
A cube-shaped box of inside width {{{2r}}} has an inside volume of
{{{V(box)=(2r)^3=8r^3}}}
The ratio of the volumes is
{{{V(ball)/V(box)=(4/3)*pi*r^3/8r^3}}} --> {{{V(ball)/V(box)=(4/3)*pi/8}}} --> {{{V(ball)/V(box)=highlight(pi/6)}}}
The sizes of ball and box do not matter.
As long as the ball fits tightly in the box, and the box is cube-shaped, the ratio is the same.
 
NOTE:
If you are curious, {{{2744mm^3=(14mm)^3}}} and that would make the diameter of your billiard ball 14mm.
That is way too small, marble size.
I did not use a calculator to find
{{{root(3,2744)=14}}}
I just divided 2744 by 2, by 2, by 2, and by 7 to get 49 and realize that the prime factorization is
{{{2744=2^3*7^3}}}
so I knew that {{{2744=2^3*7^3=(2*17)^3=14^3}}}