Turning Surface Finish from Feed and Tool Nose Radius Calculator

Estimate the ideal turning profile from feed and tool nose radius, or solve for the feed or radius needed to reach a theoretical roughness target.

Geometry and model

Theoretical Rₐ

1.5625µm

Approximate Rₐ1.5625 µm

Theoretical Rₜ6.25 µm

Feed / radius0.25

The result is an ideal geometric prediction; verify the manufactured surface by measurement.

Feed sensitivity

How to use

Select Calculate roughness, Solve feed, or Solve radius.
Choose the Rₐ approximation, approximate Rₜ, or exact circular-geometry Rₜ model.
Enter the required positive inputs and select their units. Feed is per revolution; radius is the tool nose radius.
Select Calculate. Use Reset to restore the example and the roughness mode.
Read the single primary result, supporting Rₐ/Rₜ metrics, warning, and feed-sensitivity plot. Correct any red input warning before using a value. Exact inverse-radius operation requires 0 < Rₜ < f/2; targets at or above half the feed are not on the implemented circular-profile branch and are rejected.
Treat the output as an ideal geometric estimate. Confirm actual surface texture by measurement.
Do not use this model for acceptance where vibration, chatter, tool wear, built-up edge, material deformation, runout, edge honing, or filtering dominates the profile.

Equations used

The circular nose geometry and its common small-feed approximations are:

Rₜ,exact = r − √(r² − (f/2)²)

Rₜ,approx = f² / (8r)

Rₐ,approx = f² / (32r) = Rₜ,approx / 4

Inverse exact-geometry forms are:

f = 2√(2rRₜ − Rₜ²)

r = (Rₜ² + f²/4) / (2Rₜ)

Symbols and units. f is feed per revolution (mm/rev or in/rev); r is tool nose radius (mm or in); Rₜ is theoretical peak-to-valley cusp height (µm or µin); Rₐ is the arithmetic-average roughness approximation (µm or µin). All lengths are converted to consistent units internally. Only positive geometric heights are reported; no signed profile direction is implied.

Assumptions and limits. The tool has a circular nose and the pass is rigid and kinematically ideal. Exact forward geometry requires 0 < f < 2r, exact inverse-feed operation requires 0 < Rₜ < r, and exact inverse-radius operation requires 0 < Rₜ < f/2. At Rₜ ≥ f/2, algebraic squaring selects the other circular branch, so the target is rejected. The approximations are best when f is small relative to r. Actual texture can differ because of vibration, chatter, tool wear, built-up edge, material response, runout, edge preparation, and measurement filtering.

References. Geometry background: Engineers Edge — Cutting Tool Radius and Surface Finish. Surface-texture terminology: ISO 21920-2:2021, which supersedes ISO 4287 terminology.