Friday, October 23, 2015

3D Printing: How Long is a Roll of PLA Filament?



I've been wondering for a while how long a roll of 3mm PLA is, and today I decided to find out. Not by unrolling the filament roll, of course, but by using a little math.

First, we should find the volume of the filament, and then we should find the height of an ideal filament "cylinder"- a straight, unrolled filament roll.

We can determine the filament density quite easily. NatureWorks LLC provides much of the American PLA supply and is a good indicator of the density of general PLA filaments. Although there are many different PLA formulations, all with different densities, the most common formulation is NatureWorks' Ingeo 4043D.

Luckily, NatureWorks provides an online PDF with the specifications for Ingeo 4043D.


And density is in the PDF (see top line in above image)- it is 1.24 g/cc (grams per cubic centimeter). Now we can find how long the filament roll is.

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We can reformulate the density formula, knowing the mass and density of the filament roll.


The mass of the filament roll is 1 kg, or 1000 g. The density (already provided) is 1.24 g/cm^3.


Now that we know the mass and density of the filament roll, we simply substitute values into the density equation to obtain the volume.


This looks like a good value. Now, assuming that the unrolled filament line is a cylinder, with the top and bottom circles with 3 mm diameters, and a long length, we can calculate the length of the cylinder.

We know volume in the cylinder, pi, and the radius. To solve for height, we simply reformulate the equation.


We can fill in the radius of the filament cylinder as 3.00 mm / 2, or 0.15 cm. The volume was provided from an earlier step, and pi is defined.


Now we can simply substitute values into the height-volume-radius equation and solve.


Looks good... let's convert to meters.


Using significant figure notation, a roll of 3D printer filament is about 114 m long. Obviously, this number will vary based on the accuracy of the filament-extruding machine, the accuracy of the scale that weighs the PLA, and the formulation of PLA; however, this is a an accurate answer that should cover most questions about the length of a roll of PLA.

Interesting stuff! I'll be back next week!