Ask A Scientist Mathematics Archive

Question: Is there any mathematical formula for determining the un-rolled 
length of a spiral?  For instance, if I had a roll of ribbon that had a radius 
of two inches, and the ribbon was one sixteenth of an inch thick, how long 
would the ribbon be?  If there is not a formula for it, what would be the best 
way to determine this? 
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A very close approximation for this kind of spiral is to pretend 
that the roll of ribbon is a series of concentric  circular loops of ribbon.  
Then, the length of each loop is 2 * pi * r, where r = the loop's radius and 
pi = 3.1415926...so the answer for your example would be:  2 * pi * (1/16 + 
2/16 + 3/16 + ... + 32/16) = 2 * pi * 528/16.  If the ribbon was on a spindle 
of radius, say, 1 inch, the sum would be (16/16 + 17/16 + ... + 32/16).  An 
interesting related problem is find (and prove) a formula for ( 1 + 2 + 3 + 
... + N ) 
John Hawley
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Here is another approach.  If we assume that the ribbon is tightly 
wound up (i.e., no gaps between the layers of ribbon), then the volume 
occupied by the 'disk' of wound-up ribbon equals the volume of the unwound 
strip (a very flat and long 'box').  Let R denote the radius of the 'disk', T 
the thickness of the ribbon, L its unwound length (the quantity we want to 
compute), and W its width.  The wound-up volume is pi*R^2*W ("R^2 means "R 
squared"), and the unwound volume is L*W*T.  Setting these equal and solving 
for L, we get L = pi*R^2/T.  For the values given (R=2 in., T=1/16 in.), we 
get L = 64*pi (about 201) inches.  By the way, your idea of using a spiral 
would work.  One could write an equation (polar coordinates would be the 
easiest choice) for a curve that would lie within the ribbon-say, at mid-
interior.  There is a formula for length of a curve, using integral calculus.  
As it happens, the answer you get is the same as that obtained above.  If you 
would like more details on this calculus approach, please ask (here or via e-
mail).
Ron Winther
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