Staircase Surface Roughness

A sloped surface built from flat layers is a staircase, and there is no way around it — it is the defining geometric compromise of layered manufacture. The steps have a rise equal to the layer thickness $t$ and a tread of $t/\tan\theta$, where $\theta$ is the angle the surface makes with the build plate.

Getting from that to a roughness takes one piece of geometry. Measured perpendicular to the nominal sloped surface, the profile deviates from the ideal plane by at most $t\cos\theta$ — you can check it on a 45° face, where the step is a square of side $t$, the ideal surface is its diagonal, and the corner stands $t/\sqrt{2}$ off that diagonal. The deviation profile is a triangular wave, and the arithmetic mean deviation of a triangular wave is a quarter of its height. Hence $R_a \approx (t/4)\cos\theta$, which is the relation Byun and Lee used in their 2006 work in the International Journal of Advanced Manufacturing Technology on choosing build orientation.

Two limits are worth feeling. A vertical wall at 90° gets zero, because a vertical face has no steps — each layer sits directly on the one below. A shallow slope approaches $t/4$ and no worse, because however long the treads become, the deviation measured normal to the surface can never exceed one layer thickness. And a truly flat top at exactly 0° has no steps either, so the formula's limit of $t/4$ there is a degenerate case where the model stops applying rather than a prediction.

Some references quote this with a cotangent instead, $(t/4)\cot\theta$, which is a quarter of the horizontal tread width rather than the deviation normal to the surface. The two agree closely on steep faces — at 80° they differ by about one per cent — and diverge badly as the surface flattens, where the cotangent form runs off to infinity while the real deviation is capped at one layer. If you are comparing against a published figure, check which form it used before concluding anything.

Now the part that matters most, and it is not in the equation at all. This is geometry only, and on a metal powder-bed part the real surface is usually dominated by something else entirely: partially melted particles sintered onto the surface by heat leaking out of the melt pool. Those particles are 15 to 45 µm across — the same size as, or larger than, everything this formula computes. Measured as-built $R_a$ on laser powder-bed parts typically runs 8 to 20 µm on upskin faces against staircase predictions of a few micrometres, and downskin surfaces are worse again, because they are melted against loose powder rather than solid metal, so the pool sinks, drags particles with it, and forms dross. The staircase relation is a floor, not a prediction.

Where it genuinely does dominate is fused filament fabrication, and that is worth saying plainly, because it is the machine most readers own. Layers there are 0.1 to 0.3 mm rather than 0.03, ten to twenty times thicker, and nothing is sintering onto the surface — so the steps really are what you see and what you feel, and $(t/4)\cos\theta$ is a fair description of a filament part's sloped face.

Used honestly, the relation is a comparison tool for orientation rather than a value for a drawing. Solving it for the angle turns it into a rule — this is the shallowest a face may be laid at, on staircase grounds — which is one input among several: support volume, build height and so build time, residual stress, and which faces will be machined afterwards. If the answer demands a layer thinner than the machine is qualified for, the honest conclusion is that the surface must be machined, tumbled, blasted or chemically polished after the build. That is very often the right answer, and much cheaper than doubling the build time chasing a finish the process cannot give.