Calculating the Diameter of a Carpet Roll

Date: 9/24/95 at 12:36:46
From: Anonymous
Subject: Calculating the diameter of a roll

How do you calculate the diameter of a carpet roll 
when you have the length and the thickness?

Thanks in anticipation.

Filip 

Date: 9/24/95 at 18:36:18
From: Doctor Andrew
Subject: Re: Calculating the diameter of a roll

Hi!  Great question.  For a darn good approximation you could 
find the cross-sectional area (the area of the circular end of 
the roll) needed for the roll, assume that the roll is a 
perfect cylinder, and then calculate the diameter required for 
that cross-sectional area.  If a carpet has length l and 
thickness t, the cross-sectional area of the roll is the same 
as the cross-sectional area of the edge of the carpet when it 
is lying on the floor, which is the very long rectangle with 
length l and thickness t.  See if you can take it from here to 
find the diameter.

Neither of us has ever seen a carpet roll shaped perfectly 
cylindrically since it is a spiral with an edge that usually 
sticks out off the rest of the roll.  I imagine that trying to 
describe the shape of the spiral could be pretty tough.  You 
could probably come up with a number of approximations, the 
most accurate of which would take into account how the spiral 
begins and what kind of space would be left in the center of 
the roll due to physical limitations on how the carpet can 
bend.  You might notice that the approximation in the above 
paragraph becomes less and less accurate as the ratio of l to t 
(the fraction l/t) becomes smaller and smaller.  If you think 
more about it, you'll realize that when a carpet bends it is 
already changing its shape (stretching on the outside, 
compressing on the inside) in a way that makes describing it 
exactly with mathematics very difficult.

Thanks for your question.  If you need more help, please write 
back.

-Doctor Andrew, The Geometry Forum