Length of Material on a Cylindrical Roll

Date: 6/21/96 at 21:56:30
From: Anonymous
Subject: Length of Material on a Cylindrical Roll

Is there a formula for calculating the length of material on a roll
knowing the roll diameter, the core diameter, and the material
thickness?

I recently tried using a formula I saw in a newsletter, but it was
not even close when I tried a real life example.

The formula was something like (64.54*(D*D)-(C*C))/T*1000 where D is
the roll diameter, C is the core diamter, and T is the thickness, all
in inches.  The answer was supposedly in feet.

Thanks.

Date: 6/22/96 at 11:37:6
From: Doctor Anthony
Subject: Re: calculating material length on roll

The area of cross-section of the roll can be expressed in two ways.  
When stretched out in a straight line it is a rectangle of area L*T 
where L is the length in inches, and T the thickness viewed edge on.  
When rolled on the drum, this same cross-section will be pi*R^2 - pi*
r^2 where R = outer radius =(D/2), and r = inner radius (=C/2)

So we have L*T = pi{D^2/4 - C^2/4}

             L = pi/(4T){D^2 - C^2}  inches

        Length = pi/(48T){D^2 - C^2} feet 

               = (0.06545/T){D^2 - C^2) feet  (T,D and C in inches)

Try this in real life.  It will be sensitive to the value of T you 
use, so measure that carefully.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/