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.
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.
Now let's have a look on how these formulas are deduces.
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When you are measuring your rolls to get the sizing to pass along either to a Sales Representative or to find your paper on the website, you need to follow the picture below.
You need to measure the width of the receipt from one end to the other, then the diameter of the roll on the side, and you may need to measure the inner or outer core depending on your machine.Let D the outer diameter of the roll, h the thickness of the tape and N the number of turns. 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 2(N–1)h).The total length of the tape is therefore: 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).If all conditions are met the shelf life of the paper is up to seven years.As well once the thermal paper has been used the image life of the paper is up to seven years as long as all storage conditions are met.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.Standard inner core size is 7/16 inches and normally the thermal coating is on the outside of the roll.With some ATM rolls and Cash Register rolls requirements maybe different like thermal coating on the inside of the rolls, a 1 inch inner core, and etc., any and all specifications will be on product’s information on the website.The difference is only 157 mm over 235 m, an error of 0.067 %.If we consider that when we measure the diameter the precision is probably about 1 mm over 1 m (0.1 %) the error of the approximate formula is small.