Determining Length of Material Remaining on a Roll

Date: 11/24/2003 at 11:40:51
From: Laura
Subject: Need formula for determining linear length of material

Many apologies if this is not what you do, but I'm at a loss for where 
to find this answer.

I need to know the mathematical formula to determine the length of 
material as it is on a roll.  Specifically; my company supplies rubber 
and tape, both in rolls on a core.   Without unrolling the roll, we'd 
like to determine the length of the material with the following 
information:   Outside diameter of the core, outside diameter of the 
whole roll and thickness of the material (determined by a micrometer).   

Do you have any such formula?

Laura

Date: 11/24/2003 at 15:51:25
From: Doctor Peterson
Subject: Re: Need formula for determining linear length of material

Hi, Laura.

We get this question a lot; here are some of the explanations in our 
archive:

  Length of Material on a Cylindrical Roll
    http://mathforum.org/library/drmath/view/51723.html 

  How Much Carpet is Left?
    http://mathforum.org/library/drmath/view/56481.html 

  Fabric Left on a Roll
    http://mathforum.org/library/drmath/view/54357.html 

Briefly, the length of the material will be the product of the number 
of layers and the average length of one layer in the roll. Both are 
easy to find:

  Number of layers = (Do-Di)/(2t) [total thickness/one layer]

  Average layer = pi(Do+Di)/2     [circumference at average diameter]

where Do and Di are the outer and inner diameters, and t is the 
thickness of the material.

So the length is

  L = pi(Do+Di)(Do-Di)/(4t)

    = pi/4 (Do^2 - Di^2)/t

A small error in the thickness measurement can make a big difference; 
I would recommend measuring it not with a micrometer, but by measuring 
the outside diameter of a roll of known length and calculating t from 
this formula in reverse. That will ensure that the number you use 
reflects the way the material lies on the roll. But try both 
measurements to see how they compare.

If you have any further questions, feel free to write back.

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