Box for a Ball

Date: 09/26/2002 at 23:45:08
From: Alfreda 
Subject: Geometry

The volume of a ball is 36pi cm^3. How do I find the dimensions of a 
rectangular box that is just large enough to hold the ball?

Date: 09/30/2002 at 21:55:57
From: Doctor Ricca
Subject: Re: Geometry

Alfreda -

First, let's think about what you're really asking.  If we want a 
rectangular box that's just large enough to hold a ball, then we know 
that (1) that box is actually a cube (since a sphere is symmetric in 
every direction, there's no reason one side of the box should need to 
be longer than the others), and (2) the distance across the box (which 
is the length of any one of its edges) should be the same as the 
largest distance across the sphere (which is its diameter).

Now all we need to do is find the diameter of the ball in question, 
and then find the volume of the box with an edge length equal to that 
diameter.

The volume of a sphere can be written

    V = (4/3)*pi*(r^3)

where r is the radius of the sphere. Since you already know the volume 
of the sphere, you can solve for the radius by doing a little bit of 
algebra. Then you can find the diameter by doubling the radius.

Once you know the diameter of the ball, you can find the volume of the 
box. For a cube,

     V = L^3

where L is the length of one of its edges - which, in this case, is 
just the diameter of the sphere.  

Here's some more information about cubes and other boxes, from the Dr. 
Math Geometric Formulas FAQ:

   Rectangular Parallelepiped, Cube Formulas
   http://mathforum.org/dr.math/faq/formulas/faq.parallelepiped.html 

and some information about spheres and similar shapes:

   Sphere Formulas 
   http://mathforum.org/dr.math/faq/formulas/faq.sphere.html 

   Ellipsoid, Torus, Spherical Polygon Formulas
   http://mathforum.org/dr.math/faq/formulas/faq.ellipsoid.html 

- Doctor Ricca, The Math Forum
  http://mathforum.org/dr.math/