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Taping a Cylinder

Date: 01/29/2001 at 05:28:37
From: Gerald Robinson
Subject: Helixes

If I want to wrap sticky tape around a cylinder to cover it, what is 
the relation between the diameter of the cylinder, the thickness of 
the tape, and the angle between the diameter of the cylinder and the 
length of the tape?

Date: 01/29/2001 at 12:31:33
From: Doctor Rob
Subject: Re: Helixes

Thanks for writing to Ask Dr. Math, Gerald.

I drew the following diagram of the cylinder sliced up one side PQ
and unrolled:

   p'                             U'  Q'
    |\         \         \         \ |
    |A\         \         \         \|
    |  \         \         \       _,o T
    |   \         \         \  _,-'  |
    |    \         \       S o'      | Pi*d
    |     \         \         \      |
    |      \         \         \     |
    |   w _,o R       \         \    |
    | _,-'   \         \         \   |
   P               h              U  Q

The relation between the width of the tape w, the diameter of the
cylinder d, and the angle A is given by

   sin(A) = w/(d*Pi),

        A = arcsin(w/[d*Pi]).

The total area of the tape must equal the lateral area of the
cylinder, which is

   Pi*d*h = L*w,

if you don't waste any tape (possibly by cutting a triangular piece
PP'R off at the beginning and using it at the end to make UQTS and
TQ'U'). Here L is the length of the piece of tape, and h is the height 
of the cylinder. Thus

   L = Pi*d*h/w.

Probably you would waste the pieces adjacent to PP' and QQ' whose
total area is PR*P'R = w*sqrt(Pi^2*d^2-w^2), using one continuous
piece of tape, so that

   Pi*d*h = L*w - w*sqrt(Pi^2*d^2-w^2),

          = w*(L - Pi*d*cos(A)),

   L = Pi*d*h/w + sqrt(pi^2*d^2-w^2).

If you need further assistance, write again.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry

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