Knowing the length of a roll just by measuring its diameter can be useful in many situations, since many everyday objects are in rolled form, like tapes, paper, plastic films, and so on.

There are two ways of calculating the length of roll (mathematically speaking a "spiral"), an exact and complex formula derived from integral calculation and an approximate and simpler formula derived from the sum of circumferences of concentric circles. The approximate formula is enough in many situations as long as the thickness of the tape is small compared to the diameter of the roll.

The following calculator will simplify the math and will do the calculations for you (with the exact or the simplified formula):

Mobile version available here.

Now let's have a look on how these formulas are deduces. To keep things simple, let's consider the spiral as a series of concentric circles (one per turn) where the radius increases by the thickness of the tape every turn. The error will be small if the thickness is small compared to the minor diameter of the roll.

Let *D _{0}* be the inner diameter of the roll (the diameter of
the core),

First, the number of turns is easy to compute, since it's just the difference of the two radii divided by the thickness of the tape or, in terms of diameter:

The circumference of the first turn is *πD _{0}*, the one of
the second turn

By rearranging the terms we get:

Now, the sum of the integers from *1* to *N-1* can be expressed in
the form *N(N–1)/2* (Gauss's formula).
The length *L* can therefore be expressed as:

Or, in a simpler form:

Now, if we want to do the reverse operation, meaning to calculate
*D _{1}* as a function of

Second degree equations usually have two solution, expressed with a "±" sign; but in this case we just want the solution with "+", as the solution with "–" corresponds to the same spiral, but coiled in the other direction.

Than we calculate *D _{1}* with:

To calculate the exact length of the spiral, we write the equation of the curve in polar coordinates:

Here *ρ* is the distance between the axis as a function of the angle
*φ*.
*φ* is expressed in radians and is *0* at the beginning and
increases by *2π* every turn.
So, every turn the radius *ρ* increases by *h*.

To calculate *φ* we proceed as we did before to calculate *N*,
but now we have a factor *2π* because of the angle in radians:

and

The length of a curve in polar coordinates is calculated with the following formula:

Here, the derivative is easy to calculate and we get:

So, to find the exact length, one just has to calculate:

Now, the things are more complicated.
Even if a good integral solving list should include the solution of such
integral, if we want to calculate it by hand, we'll have first to change the
variable using the hyperbolic identity
*ch ^{2}x – sh^{2}x = 1* and
than transform the result with another hyperbolic identity:

Inverting this function is not easy, and in this application a numerical approximation (the Newton's method) is used. This method start with an approximation or a guess of the solution (zero of a function) and iteratively refines it by finding the intercept point on the abscise axis of its tangent line. Since this point is a better approximation, so the process can be repeated until the desired precision is achieved.

Here we have *L(φ _{0},φ_{1})=L_{0}* and
we want to find

The Newton's method tells us that a better approximation of *φ _{1}*
is

And we can iterate the above equation until the desired precision is
achieved.
The calculator in this page tries to reach as much precision as it can be
stored in a regular JavaScript floating point variable, so that no error can
be remarked.
Usually a few iterations are enough.
To start this algorithm, we can use the approximate formula to find an
initial *φ _{1}*.

In order to implement this method, we need to compute by hand the derivative
*∂f(φ _{0},φ_{1})/∂φ_{1}*,
which is just

With this the inversion algorithm can be easily implemented in the calculator.

To have an idea of the error introduced by the approximate formula, let's
take an example: suppose we have a roll of paper with an inner diameter
*D _{0} = 100 mm*, an outer diameter

The advantage of the approximate formula is that it uses only basic operations and can be easily computed even by hand on a simple pocket calculator. The exact formula uses a lot more math, needs at least a good scientific calculator, is long to do by hand and has little gain on precision. On the other hand, on a computer or a smart-phone, there is enough CPU power for this JavaScript calculator to consider the exact formula as a better option.

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