Radius of a Tennis Ball

Date: 04/03/2001 at 16:11:05
From: Mr. Goodsky's Class
Subject: Volume of a Sphere

Hi Dr. Math,

As we were working on determining how many tennis balls would fit in a 
box, we discovered that we really couldn't get the EXACT amount 
without determining how to find the volume of a sphere.

While doing research I found the formula for finding the volume of a 
sphere as  4/3 times pi times the radius cubed.

We were unable to accurately find the radius beyond a good estimate, 
so our question is: How do we find the volume of a tennis ball 
without cutting it open?

Thank you,
Mr. Goodsky's 5th grade class

Date: 04/03/2001 at 16:17:34
From: Doctor Robert
Subject: Re: Volume of a Sphere

The formula that you mention is written 4/3 * pi * r^3. You can 
calculate the volume of a tennis ball if you can determine its radius.  
It's hard to measure the radius, but it's not that hard to measure the 
diameter. What you need is a set of vernier calipers to determine the 
diameter. Of course, you divide the diameter by two to get the radius.  

I would suggest visiting your science teacher. He or she might have 
some calipers.

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

Date: 04/03/2001 at 16:40:04
From: Doctor Peterson
Subject: Re: Area of a Sphere

Hi, Class.

It's worth noticing that knowing the volume of each ball will not be 
enough. There will be a lot of extra space BETWEEN the balls, and 
that can be very hard to measure - in fact, it will depend on how the 
balls touch inside the box.

To answer your question, there are two good ways to find the radius. 
First, you can use any of various measuring devices that are designed 
to measure the thickness of round objects. One kind would have a pair 
of parallel "jaws" that can fit around the ball, and a scale to show 
how far apart the jaws are. Another kind (calipers) consists of a pair 
of points you can fit around the ball, and then put against a ruler to 
measure how far apart they are. Maybe your school has some, or one of 
your fathers.

If you can't find any thing like that, you can make something 
yourself. Just get three rulers, and place them like this:

        |                               |
     ---+-------------------------------+----
        |           *********           |
        |         **         **         |
        |       **             **       |
        |      *                 *      |
        |      *                 *      |
      4"|     *                   *     |4"
        |     *                   *     |
        |      *                 *      |
        |      *                 *      |
        |       **             **       |
        |         **         **         |
    ----+-----------*********-----------+-----

That is, put one ruler horizontally across the ball, and measure how 
far it is from the table using the other two rulers held vertically. 
If they read the same distance, then that is also the diameter of the 
ball.

Another way to find the radius is to use a flexible tape measure or 
string around the ball to measure its circumference. Divide that by 
pi, and you have the diameter.

(You might also be able to do a Web search for sites that tell the 
official size of a tennis ball.)

Try using both methods and see if the answers agree.

Now, back to the box problem. The best I can do there is to mention 
that it has been found mathematically that spheres take up no more 
than 75% at most of the space in any container, when they are packed 
in carefully; if they are randomly packed, you can expect more like 
64%. (If the balls are not a lot smaller than the box, it probably 
won't be close to this.) See if you can use that to make a good 
estimate of the number of balls you can fit.

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