Surface Finish Calculator (Ra from Feed)
Work out theoretical surface finish from feed and nose radius, or the feed a target Ra needs, with conversion between Ra, Rz, RMS and ISO N grades.
How to use this calculator
- 1Work in the direction the job demands: from a feed you plan to run, or backwards from a finish on a drawing.
- 2Leave a practical allowance of about two when working backwards. The theoretical figure is a floor and real surfaces do not reach it.
- 3Try a larger nose radius before reducing the feed. Roughness goes as the square of feed but only inversely with radius, so the radius is the cheaper lever.
- 4For anything under about 16 µin, plan on grinding or a wiper insert rather than a conventional finishing pass.
- 5Treat the number as a target to verify, not a specification met. Surface finish is measured, not calculated.
How the calculation works
Rz = f² / (8 r) peak to valley of the cusp
Ra = 0.2566 x Rz = f² / (31.2 r) mean deviation of the same profile
RMS = 1.11 x Ra the old US convention
f = sqrt(Ra x 8 r / 0.2566) feed for a target finish- f
- Feed per revolution. Finish goes as its square, so halving the feed quarters the roughness
- r
- Tool nose radius. Roughness is inversely proportional to it, so doubling the radius halves the roughness
- 0.2566
- The exact ratio of Ra to Rz for a parabolic cusp - 2 times the integral of |u² - 1/3| from 0 to 1
The 31.2 in the familiar Ra formula is 8 divided by 0.2566. It is not an independent constant; the two relations are the same geometry.
The Ra to Rz ratio of 0.2566 holds for the theoretical cusp only. Measured surfaces have ratios anywhere between 1:4 and 1:7, which is why converting a measured Ra to Rz is unreliable.
RMS at 1.11 times Ra is the usual figure for a machined surface with a roughly sinusoidal profile. It is an approximation, and old drawings marked in RMS should be treated as approximately equal in number to Ra.
Worked example
A normal finishing pass
- 1.0.008 in/rev on a 1/32 in nose radius.
- 2.Rz = 0.008² / (8 x 0.03125) = 0.000064 / 0.25 = 0.000256 in, which is 256 µin.
- 3.Ra = 0.2566 x 256 = 65.7 µin, or 1.669 µm.
- 4.That just meets N8 at 125 µin, and would not meet N7 at 63 - in practice, with real surfaces running around twice theoretical, it will not meet N8 reliably either.
Result: 65.7 µin Ra - N8 on paper, marginal in practice
Working back from a 63 µin drawing callout
- 1.A 63 µin requirement with a 2x allowance means aiming at 31.5 µin theoretical.
- 2.f = sqrt(31.5e-6 x 8 x 0.03125 / 0.2566) = sqrt(3.068e-5) = 0.00554 in/rev.
- 3.So about 0.0055 in/rev rather than the 0.008 that would give 63 on paper.
- 4.The difference is the whole point: feeding to the theoretical figure is how a part fails inspection on a dimension nobody thought was tight.
Result: 0.0055 in/rev - not the 0.0078 the arithmetic alone suggests
The same finish with a bigger nose radius
- 1.Doubling the nose radius to 1/16 in, aiming at the same 31.5 µin theoretical.
- 2.f = sqrt(31.5e-6 x 8 x 0.0625 / 0.2566) = 0.00783 in/rev.
- 3.The feed rises by a factor of sqrt(2) - about 41% more metal per revolution for exactly the same finish.
- 4.Nothing was given up except some edge engagement, which only matters on a slender part where the extra radial push would cause chatter.
Result: 0.0078 in/rev - 41% more feed for the same finish
Converting an old RMS callout
- 1.An old drawing marked 125 RMS.
- 2.RMS runs about 11% above Ra on a machined surface, so Ra = 125 / 1.11 = 112.6 µin.
- 3.That is 2.86 µm, and it meets N8 at 125 µin - comfortably rather than marginally.
- 4.Treating 125 RMS as 125 Ra would be slightly conservative, which is the safe direction, but it is not the same number.
Result: 125 RMS is about 113 µin Ra - grade N8
Where the cusps come from
A turning tool does not have a straight edge at the point of cut; it has a nose radius, typically a sixty-fourth to a sixteenth of an inch. As the tool feeds along, that radius sweeps a path, and because the tool moves sideways by the feed every revolution, consecutive sweeps overlap only partly.
What is left between them is a ridge - a cusp - running as a helix along the part. Its height is pure geometry: two circular arcs of radius r whose centres are a distance f apart intersect at a height very close to f²/(8r) above the deepest point, when f is small compared with r.
That is the peak to valley roughness, Rz. It sets the ceiling on what any finishing pass can achieve, and it explains the two levers immediately: feed enters as a square, so halving it quarters the roughness, while radius enters linearly, so doubling it halves the roughness.
Deriving Ra rather than asserting it
Ra is not the peak to valley height. It is the mean absolute deviation of the profile from its own mean line, which is a different and smaller number, and the relation between the two is usually just quoted.
It can be derived exactly. Approximate the cusp as a parabola - accurate when the feed is small relative to the radius - and put it in normalised coordinates where the profile is y = Rz x u² across u from -1 to 1. The mean line sits at the average of that, which is Rz/3. The profile crosses its own mean line where u² = 1/3, at u = ±0.5774.
Integrating the absolute deviation over the period gives 2 times the integral of |u² - 1/3| from 0 to 1, which evaluates to 0.2566. So Ra = 0.2566 x Rz, exactly, for this profile shape.
Substituting back gives Ra = 0.2566 x f²/(8r) = f²/(31.18 r), which is the familiar formula printed as Ra = f²/(31.2 r) everywhere. The two relations are not independent facts to be looked up; they are the same geometry expressed twice.
One important restriction follows. The 0.2566 belongs to the theoretical cusp. A measured surface has tool marks, tearing, built-up edge fragments and vibration on top of the cusps, and its Ra to Rz ratio can be anywhere from about 1:4 to 1:7. Converting a measured Ra into a measured Rz is genuinely unreliable, and no standard sanctions it.
Why the real surface is always worse
The formula gives the best possible outcome for a given feed and radius. Actual turned surfaces come out roughly one and a half to three times rougher, and it is worth knowing what the extra consists of, because each part has a different remedy.
Built-up edge is usually the largest contributor and the most fixable. At low cutting speeds, workpiece material welds to the tool face, grows, and periodically breaks away, taking some of the surface with it and leaving the tool a different shape than it started. It is worst in gummy materials - low carbon steel, aluminium, copper - at moderate speeds, and the cure is more speed, a sharper and more positive edge, or a coating the material will not adhere to.
Vibration is next. Any relative movement between tool and part is written directly into the surface, and a finishing pass is where the cut is lightest and the system is least damped. Chatter marks are unmistakable when severe, but low-level vibration just makes everything slightly rougher with no obvious signature.
Then tool wear, which changes the effective nose radius during the cut - usually for the better at first as the edge hones itself, then sharply worse. And finally the material: below a critical cutting speed the chip tears rather than shearing, and no amount of feed adjustment fixes a torn surface.
The practical consequence is simple. If the theoretical figure comes out just inside the requirement, the requirement will not be met. A factor of about two is a sensible planning allowance.
Wipers, and when the formula stops applying
A wiper insert has, behind the main nose radius, a series of larger blended radii arranged so that as the tool feeds along, the trailing part of the nose passes back over the cusps the leading part left and flattens them.
The effect is to break the relationship on this page in a useful direction. A wiper produces roughly the same finish at double the feed, or roughly half the roughness at the same feed. Where a finishing pass is the bottleneck in a cycle it is often the cheapest available improvement, and it costs only a little more per insert.
The trade-offs are real but narrow. The larger effective contact pushes harder radially, which slender parts do not like. Wipers want a reasonably rigid setup and a consistent depth of cut, and they are less tolerant of interrupted cuts. And on a profile with tight corners the geometry cannot be used at all.
The formula also stops describing anything once the feed approaches the nose radius. At that point the passes barely overlap, the parabolic approximation fails, and the surface is a series of separate grooves rather than a cusped profile. That is a roughing cut, and its finish is not the point.
What this assumes, and where it stops
Assumptions
- A round-nosed single point tool in turning or boring, feeding along the part.
- The feed is small relative to the nose radius, so the cusp can be treated as parabolic.
- Ra = 0.2566 x Rz applies to this theoretical cusp shape, not to measured surfaces.
- RMS at 1.11 times Ra assumes a roughly sinusoidal profile, which is the usual machined case.
- No built-up edge, vibration or tool wear - all of which make the real surface rougher.
Limitations
- It gives a theoretical floor. Real surfaces run one and a half to three times rougher.
- It does not describe wiper inserts, which deliberately flatten the cusps and roughly double the usable feed.
- It does not cover milling, where the surface is generated by a rotating cutter and the geometry is different.
- Converting between measured Ra and measured Rz is unreliable regardless of what this page computes for the theoretical case.
- Cutting speed, coolant, coating, rake and material condition all affect real finish and none are modelled.
Common questions
What is the formula for surface finish from feed rate?
Ra = f² / (31.2 x r) with feed and nose radius in inches, giving Ra in inches. The peak to valley form is Rz = f² / (8r), and Ra is exactly 0.2566 of that - which is where the 31.2 comes from, since 8 divided by 0.2566 is 31.18.
How do I convert Ra to Rz?
For the theoretical cusp, Rz = Ra / 0.2566, so about 3.9 times. For a measured surface there is no reliable conversion - real ratios range from about 4 to 7 depending on the process and what else is affecting the surface. No standard endorses converting measured values.
What is the difference between Ra and RMS?
Ra is the mean absolute deviation from the mean line; RMS is the root mean square of the same deviations, which weights peaks more heavily. On a machined surface RMS runs about 11% above Ra. Old US drawings marked in RMS are close enough to Ra numerically that treating them as equal is slightly conservative.
How do I get a better surface finish when turning?
Increase the nose radius before reducing the feed - roughness is inversely proportional to radius but proportional to the square of feed, so a bigger radius costs nothing in cycle time. Then raise the cutting speed to get clear of built-up edge, make sure the edge is sharp, and tighten up the setup. A wiper insert doubles the usable feed for the same finish.
Why is my actual finish worse than the calculation?
Because the calculation is a geometric floor and everything else adds to it. Built-up edge is usually the biggest contributor, especially in gummy materials at moderate speed. Then vibration, tool wear changing the effective radius mid-cut, and the material tearing rather than shearing at low speed. Plan on one and a half to three times theoretical.
Sources
- Turning formulas and definitions - surface finish — Sandvik Coromant
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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