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### 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

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/
```
Associated Topics:
College Euclidean Geometry
College Geometry
High School Euclidean/Plane Geometry
High School Geometry
High School Practical Geometry

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