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Finding the Length of Carpet Left on a Roll

Date: 09/25/2008 at 20:08:45
From: Lee
Subject: Can you find the total square yardage of a roll of carpet

I can see the end of a roll of carpet and figure out the approximate 
square yardage by taking the inside circumference, outside 
circumference and number of total layers left.  I need to see if 
there is a formula.  

EX: 12"=inside roll circumference, 24"=outside roll circumference, 
12 layers on end of roll.  The difference between 24" and 12" is 
12", so each layer must be about 1" more than the previous layer 
(i.e. first layer length = 12", 2nd = 13", 3rd = 14" . . . 24th = 
24").  Now, when I add all layers together I get a certain total 
length.  What is the equation for such a problem?

Date: 09/25/2008 at 23:41:06
From: Doctor Peterson
Subject: Re: Can you find the total square yardage of a roll of carpet

We have many answers to this kind of question; you might try searching
our archives for the phrase "roll of carpet" to find those that happen
to be about carpets.

Here is one, with references to others:

  Determining Length of Material Remaining on a Roll 

My formula there is

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


  Do = outer diameter
  Di = inner diameter
   t = thickness of one layer

You give the circumferences, which we'll have to divide by pi, and
since you know the number of layers, we can make a different formula
based on a number of layers N.

Let's say we have

  Co = outer circumference [so Do = Co/pi]
  Ci = core circumference  [so Di = Ci/pi]
   N = number of layers    [so t = (Do-Di)/(2N) = (Co-Ci)/(2 pi N)]

Then my formula becomes

  L = pi/4 (Co^2 - Ci^2)/(pi^2) * (2 pi N)/(Co - Ci)
    = (Co + Ci)*N/2

For your numbers this gives

  L = (24 + 12)*12/2 = 216 inches

This is exactly what I would get as an approximation using your idea
of the linear increase in circumference with each layer.  The sum of
an arithmetic series like your 12 + 13 + 14 + ... + 23 (note that you
counted 13 layers by including both 12 and 24) is the average of the
first and last, times the number of terms.  In your example, this is
12*(12+23)/2 = 210 inches.  But what about the last layer?  We only
got to a length of 23.  If we split the difference and treat the
series as 12.5 + 13.5 + ... + 23.5, our length turns out to be
12*(12.5+23.5)/2 = 216.  And in fact, we could derive the formula
above starting with this idea of the arithmetic series.

Actually, I suspect that in the case of carpet it is possible that the
backing would be more rigid than the top, so that the actual length
would be better estimated by the 210 inch figure; my 216 assumes that
the outside of each layer expands and the inside contracts equally
when it is rolled up.  The formula using your idea turns out to be

  L = (Co + Ci - (Co - Ci)/N)*N/2
    = ((N-1)Co + (N+1)Ci)/2

I've never looked at it from this perspective before and obtained this
particular formula!  It's not quite as simple, and considering that
the carpet really forms a spiral rather than neat layers, it's still
just an approximation; but it has an interesting "elegance"!

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

- Doctor Peterson, The Math Forum  
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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