Where a machined finish actually comes from
A single-point tool with a radius on its nose does not cut a flat surface. It cuts a series of overlapping arcs, one per revolution, and between each pair of arcs a small ridge of uncut material survives. Those ridges are the surface finish. Their height is fixed by how far the tool moved between passes and by how tightly curved the nose is.
That gives a purely geometric prediction, and it is a good one for turning and for face milling with a round-cornered insert. Feed enters squared and radius enters linearly, which is the practical point: halving the feed quarters the roughness, while doubling the nose radius only halves it. When a part comes off too rough, feed is almost always the cheaper lever.
What the geometry cannot predict is everything else — built-up edge, a worn corner, chatter, a workpiece that flexes. Those only ever make the finish worse than the theory, never better. Treat the number here as the best case the setup can achieve.
Ra, Rz and RMS are three different questions
Ra is the arithmetic average deviation from the mean line — add up how far the profile strays from centre, ignore the sign, and average it. It is the number on almost every modern drawing because it is stable and easy to measure.
Rz is peak-to-valley height. For the circular cusp this geometry produces, Rz works out at exactly four times Ra, which is why the two formulas differ only in whether you divide by 8 or by 32. Rz matters where a single deep scratch would matter — sealing faces and fatigue-critical surfaces.
RMS is the root-mean-square average, and it appears on older American drawings. For a machined profile it runs about 11% above Ra, so a 63 RMS callout is roughly a 57 Ra requirement. Reading an RMS number as though it were Ra makes the requirement look tighter than it is.
Worked example: 0.008 in/rev with a 1/32 nose radius
A finishing pass at eight thou per revolution, using an insert with a 1/32 inch (0.031 in) corner radius.
- Square the feed. 0.008² = 0.000064.
- Multiply the radius by 32. 32 × 0.031 = 0.992.
- Divide. 0.000064 ÷ 0.992 = 0.0000645 inch, which is 64.5 microinches Ra.
- Peak to valley. Dividing by 8 instead of 32 gives 258 microinches Rz, four times the Ra.
Now suppose the drawing calls for 32 Ra. Rearranging the formula, the feed you need is √(32 × 0.031 × 0.000032) = 0.0056 in/rev. Note that halving the roughness did not need halving the feed — it needed dividing by the square root of two, because feed is squared. Fitting a 1/16 inch radius instead would have hit 32 Ra at the original 0.008 feed, and kept the cycle time.
What finishes are realistic, and where theory stops
Ordinary turning and milling comfortably reach 63 to 125 microinches. A careful finishing pass with a sharp insert and a rigid setup gets to 32. Below about 16 microinches the theoretical calculation stops being the constraint: built-up edge, minute tool wear, spindle runout and vibration all contribute more than the cusp geometry does, and no amount of feed reduction reliably gets you there.
That is the point at which the honest answer is a different process. Grinding routinely holds 8 to 16 microinches, honing and lapping go below 4. A drawing calling for 8 Ra on a turned diameter is usually a drawing that expects the part to be ground.
There is also a floor from the feed itself. Feed per revolution has to stay above the cutting edge's minimum chip thickness, or the tool stops cutting and starts rubbing — which burnishes, work-hardens, and produces a worse finish along with rapid wear. Very light finishing feeds on a large nose radius run into this, and it is why the finish sometimes gets worse when you slow the feed down.
Common finish callouts and what they take
| Target Ra | 1/64 in (0.0156) | 1/32 in (0.031) | 1/16 in (0.062) |
|---|---|---|---|
| 125 µin | 0.0079 | 0.0111 | 0.0157 |
| 63 µin | 0.0056 | 0.0079 | 0.0112 |
| 32 µin | 0.0040 | 0.0056 | 0.0080 |
| 16 µin | 0.0028 | 0.0040 | 0.0056 |
| 8 µin | 0.0020 | 0.0028 | 0.0040 |
Feeds in inches per revolution. The 8 microinch row is included for completeness; in practice that finish is a grinding requirement rather than a turning one.
Theory is the best case, never the outcome
Every real effect pushes the finish the same way. A worn corner radius is effectively a different tool. Built-up edge tears the surface. Chatter leaves a pattern the profilometer reads as roughness regardless of the cusp height. A part that deflects under the cut leaves a taper as well as a finish problem. If a measured surface is far worse than this prediction, the cause is one of those — not the arithmetic.
Getting a better finish, in order of leverage
- Reduce the feed. It is squared in the formula, so it is the strongest single lever. Reducing feed by 30% cuts Ra roughly in half.
- Fit a larger nose radius. Halves Ra for a doubling, and costs no cycle time. The limit is chatter: a large radius means more contact length and more radial force.
- Use a wiper insert. A wiper's flat trailing geometry effectively removes the cusp, letting you keep the feed and still hit the finish. It is the standard answer when the finish requirement is fighting the cycle time.
- Check the tool before blaming the numbers. A corner that has worn is no longer the radius you entered.
- Do not feed below the minimum chip thickness. Past a certain point the edge rubs rather than cuts and the finish gets worse as you slow down.
Where this fits in a finishing pass
Surface finish is decided at the same time as everything else in the pass, and the constraints interact. The feed this page suggests still has to be reachable at a sensible spindle speed — the cutting speed calculator converts the material's surface footage into rpm, and the milling feed rate calculator turns feed per tooth into the table feed your control wants.
On a light finishing pass, radial engagement is usually small, which means chip thinning is in play — the chip load calculator shows what the tool is really seeing, and it is often less than you think, which is exactly how a finishing pass ends up rubbing. Power is rarely the limit on a finish pass, but if you are taking a heavy roughing cut first, the spindle power calculator will tell you whether the machine can do it in one go.
Terms used here
- Ra
- Arithmetic average roughness — the mean absolute deviation of the profile from its centre line. The default callout on modern drawings.
- Rz
- Average peak-to-valley height. Exactly four times Ra for the circular cusp a nose radius leaves.
- RMS
- Root-mean-square roughness, on older American drawings. Runs about 11% above Ra for a machined surface.
- Wiper insert
- An insert with small flat sections either side of the nose radius that flatten the cusp, allowing a coarser feed at the same finish.
- Minimum chip thickness
- The smallest chip an edge can actually shear. Below it the tool rubs instead of cutting, worsening finish and accelerating wear.