Engineering/How things work

How a 3D printer builds a shape

After thiswhat you will be able to doWork out the circular slice a cone needs at each height, and predict how reducing layer height changes excess plastic and the number of passes.

Questionwhat this lesson answersA printer's nozzle can only lay a flat line of plastic within one layer at a time, never a blob straight up into the air. So how does repeating that one flat move build a solid three-dimensional shape?

Not coveredwhat this lesson leaves outWe treat the shape as already sliced and the plastic as laid perfectly. We do not cover how the plastic melts and cools, how the slicing software decides each layer's path, how the first layer sticks to the bed, or what holds up an overhang.

A 3D printer has one tool: a nozzle that squeezes out a thin thread of melted plastic and drags it around a flat bed, the way you pipe a line of icing. That squeezing-out of plastic is called extrusion. The nozzle can move the thread anywhere it likes within a single flat level. What it cannot do is lay plastic in mid-air, because the thread needs something directly underneath to land on and hold it.

Raise a little, then draw again

The whole trick lives in what happens between levels. Once the nozzle finishes drawing on one flat level, it lifts by a small fixed amount and draws again, directly on top of what it just laid. That fixed step, the gap from one level to the next, is the layer height. Call it ${h}$. On a real printer ${h}$ is around $\SI{0.2}{mm}$; here we will keep it large enough to see, from $\SI{10}{mm}$ down to $\SI{0.5}{mm}$.

A solid is a stack of flat circles

Stand a cone point up, $\SI{100}{mm}$ tall with a base radius of $\SI{40}{mm}$. Cut it with a flat slice at some height ${z}$ above the bed. The cut face is a circle: wide near the bed, small near the point, and nothing at all right at the tip. Write the radius of that circle as ${r(z)}$, the outline the nozzle has to trace at height ${z}$. We define our cone by saying its radius shrinks in a straight line from base to tip,

with ${R = \SI{40}{mm}}$ and ${H = \SI{100}{mm}}$. At the bed ${r(0) = R}$, and at the top ${r(H) = 0}$.

That makes the object simple to describe. The printer draws the circle ${r(0)}$, steps up by ${h}$, draws the slightly smaller circle ${r(h)}$, steps up again, and keeps going to the tip. The cone is nothing more than that stack of circles. Each circle is one layer, the smallest thing the printer knows how to make.

The staircase you agreed to

Drag the build slider and the cone rises circle by circle. Now look at its side. It is not a smooth slope. Every layer holds a single radius for its whole height and then jumps to the next, so the side comes out as a run of little flat steps: a staircase, not a ramp. Each disc keeps the width it had at its lower edge, which is slightly too wide for the true cone anywhere above that edge, so the printed shape sits a little outside the real one the whole way up.

That overshoot is a real cost, and you can measure it. With $\SI{10}{mm}$ layers the ten stacked discs hold about 15.5% more plastic than the true cone. Make the layers finer and the steps shrink: $\SI{5}{mm}$ layers over-count by 7.6%, $\SI{2}{mm}$ by 3.0%, and $\SI{0.5}{mm}$ layers by only 0.8%. How fine the steps are is the print’s resolution, and shrinking ${h}$ buys resolution. It is not free. A $\SI{0.5}{mm}$ layer height needs 200 passes where $\SI{10}{mm}$ needed ten, so the same cone takes twenty times as long to print. Every real print picks a spot on that trade: fine enough that the steps stop mattering for the job, coarse enough to finish in a sensible time.

The shape is its slices

Watching the two views move together is the point. The circle on one side is a single slice; the solid on the other side is the whole stack of them. They are not two separate facts about the cone, they are the same fact drawn two ways. That is the reason a printer built only to draw flat can make a rounded solid at all: a solid just is its slices, and a slice is something flat you already know how to draw. The one thing the printer never gets is the smooth surface between two slices. It only ever gets flat levels and the steps between them, and it makes those steps small enough that you stop noticing.