Where the scallop comes from
Put a ball nose of radius R on a flat surface and it sweeps a circular groove. Move sideways by the stepover s and it sweeps another. Between the two, at exactly s/2 from each centre, sits a peak of material that neither pass reached. Its height above the floor of the grooves is the difference between the radius and the vertical drop of the circle at that horizontal offset, which is R minus the square root of R squared minus (s/2) squared. That is the whole derivation, and it needs nothing but the right-angle triangle formed by the radius, the horizontal offset and the depth.
Expand that expression for small s and it collapses to s squared divided by eight R. The square is the interesting part. Halve the stepover and the ridge falls to a quarter. Quarter the stepover and it falls to a sixteenth. The pass count, meanwhile, only doubles and quadruples.
The trade, in real numbers
A quarter-inch ball nose at 0.030 stepover leaves 0.903 thousandths of an inch of ridge, and covering twelve inches takes 401 passes. Halve the stepover to 0.015 and the ridge drops to 0.225 thousandths — a quarter, exactly as the square law says — while the passes go to 801 and the time doubles. Halve it again to 0.0075 and the ridge is 0.056 thousandths, at 1,601 passes.
Read that column downward and the reason finishing strategy is what it is becomes obvious. The first halving buys you a visible improvement for double the time. The second buys you something you would need a profilometer to detect, for double the time again. Somewhere in that sequence is the point where sanding for four minutes is cheaper than machining for forty, and where that point sits is a question about your material and what happens to the part next, not about the cutter.
| Stepover on a 0.25 in ball | Scallop | Passes across 12 in |
|---|---|---|
| 0.060 in | 3.653 thou | 201 |
| 0.030 in | 0.903 thou | 401 |
| 0.015 in | 0.225 thou | 801 |
| 0.0075 in | 0.056 thou | 1,601 |
The radius is the other half of the lever
The ridge falls with the radius as well, but only linearly, so going from a quarter-inch ball to a half-inch ball at the same stepover halves the scallop rather than quartering it. That still matters, because a bigger ball costs nothing in pass count — the stepover is what sets the passes, not the diameter. Where the bigger ball costs you is reach: it cannot get into an internal corner smaller than its own radius, which is usually what forces the small one back into the job for a rest pass.
What geometry does not include
Everything on this page is the ridge that a perfect machine would leave. A long thin ball nose deflects under load, and the deflection is not the same in both directions of a zigzag pattern, so a raster finish can show a ridge at twice the stepover spacing that has nothing to do with the scallop. Runout in the collet does something similar. Neither is captured here, and both are common enough that a test cut is the only honest check.
Where this sits
The stepover and pass count feed straight into the CNC router job time calculator, which is where a finishing pass turns into hours. What the extra passes do to the cutter is on the bit life and cost per part calculator. If the finish is going to be sanded rather than machined the rest of the way, the sandpaper grit progression calculator is the other half of that decision. Spindle speed for the cutter is on the router bit speed calculator.
Questions people ask
Is a scallop the same thing as a cusp?
Yes, the two words are used for the same ridge and you will see both in CAM software, sometimes in the same dialog. Some packages ask for a scallop or cusp height and work the stepover out for you, which is the second mode on this page. Others ask for the stepover as a percentage of the tool diameter, which hides the relationship completely — twenty percent of a quarter-inch ball and twenty percent of a half-inch ball give ridges that differ by a factor of two, and the percentage looks identical.
What scallop height should I aim for?
That depends entirely on what happens to the surface next, and nobody can put a number on it for you. A part that gets primed, filled and painted can carry a ridge that a clear-finished piece of hardwood cannot. A mould that will be polished wants the ridge small enough that polishing does not have to remove much, because removing material by hand is where shape goes wrong. A part that will be sanded anyway wants a ridge fine enough that the first grit can take it out in one pass. The practical method is to machine a test tile at three stepovers, take it through your actual finishing process, and see which one stops being worth the machine time.
Why is the pass count one more than the width divided by the stepover?
Because the passes are the fence posts and the stepovers are the gaps between them. Twelve inches at a three-inch stepover is four gaps and five passes, not four. The page rounds the gap count up first, since a partial gap still needs a pass at the end of it to reach the far edge, and then adds the one. On a large surface at a fine stepover the extra pass is a rounding detail. On a narrow strip it is a real fraction of the job.
Does a bull nose or corner-radius cutter follow the same formula?
For the ridge between passes on a flat surface, the radius that matters is the corner radius rather than half the tool diameter, so you can put the corner radius times two into the diameter box and the geometry holds. Where it stops holding is on any sloped or curved surface, because a bull nose contacts at a different point on the corner as the surface tilts, and the effective spacing changes with it. On flats a bull nose is far more efficient than a ball for exactly this reason — the flat bottom does the cutting and the corner only shows at the edges of the pass.
My finished surface has ridges much bigger than this page says. Why?
The likely causes are not scallop at all. Tool deflection is the usual one, especially with a long ball nose in a light spindle: the cutter pushes away from the material under load and springs back when it unloads, so the two directions of a zigzag pass cut to slightly different depths and the eye reads the alternation as coarse ridges. Runout in the collet gives the same signature. A head that is out of tram tilts the ball so that one side of every pass digs. And on figured hardwood, fibre tear-out simply overwhelms the arithmetic. A single climb-cut finishing pass in one direction only will tell you quickly whether deflection is the culprit, because the alternation disappears.