My friend Erik asked me how much it would cost to 3D print a piece of paper.

So how much would it cost you if you run out of paper but need to write down some great idea you had on something! Let's find out. This post has a lot of math so I've warned you!

Standard paper is 8.5" x 11" but 3D printers like the rest of the world generally use the metric system, measuring parts in millimeters. So first we need to convert this to millimeters.

8.5" is 215.9mm and 11" is 279.4mm

Now that we've got that out of the way, we need to calculate the volume of our part. Standard paper is 0.1mm thick. My 3D printer can handle layer sizes this small but if yours can only handle a resolution of say 0.2mm then simply double our results. typically the slicer software will choose a thicker layer height for the first layer to help the part better stick to the build plate. Our paper model is only one layer thick. So let's assume a layer thickness of .2mm.

So 215.9*279.4*.1 = 6032.246mm^3

Let's take a detour and find out not only the cost but how much filament we would need to have for our part.

Modern 3D printers now use filament with a 1.75mm diameter.

The volume of a cylinder is

V=πr^2*h

So plugging in we get 2194.425mm of filament for our paper or about 7' 3"

Alright cool, now going back to figuring out the cost:

Filament is sold in 1Kg rolls, not by length. So we need to use the density of PLA with our derived volume of our paper to compute the mass.

The density of PLA can vary slightly but going by wikipedia it is 1240 kg/m3 or 1240Kg per 1 billion mm^3. So to print a solid cubic meter you would need 1,240 spools of plastic! Since our paper's volume is measured in cubic millimeters let's convert. One cubic mm of PLA has a mass of 1.24grams or 0.00124Kg

So the volume of a spool of filament being 1Kg is (1,000,000,000mm^3)/1240Kg = 806451.612903 mm^3/Kg

So the final calculation for the cost of our piece of paper is (cost = cost of spool * object volume/volume of a 1Kg spool)

Plugging in we get cost = $30*6032.246mm^3/806451.612903 mm^3

So that ends up being $0.22 Indeed, my printing program is showing an estimated part cost of $0.19