Surface Finish from Feed & Nose Radius
Theoretical Ra and Rt from feed per revolution and nose radius in turning, or the largest feed for a target Ra — the geometric floor of the finish.
Rt from the exact arc geometry and Ra ≈ f²/(31.2r) in µm or µin, the maximum feed for a target Ra, and Ra at feeds from half to double the working point — with the reasons a measured surface is rougher.
Example: A 0.8 mm nose radius at 0.2 mm/rev gives a theoretical Rt of 6.27 µm and Ra of 1.60 µm; to hold Ra 0.8 µm the feed must drop to 0.141 mm/rev.
The finish geometry
promises — nothing more.
Where the Rt and Ra formulas come from and why a measured surface is rougher.
The profile
A single-point tool with nose radius r advancing f per revolution leaves a train of arcs. The height from a valley to a peak is exactly Rt = r − √(r² − f²/4), which for small feeds is f² ÷ (8r); the arithmetic-average roughness of that profile is about Ra ≈ f² ÷ (31.2r) — Rt ÷ 4 in the usual approximation. In µm with mm inputs, µin with inches.
Feed for a target
Solving the Ra formula backwards gives the largest feed that meets a target: f = √(31.2 · r · Ra). The feed ladder shows Ra at half to double the working feed, since Ra grows with the square of the feed: halving the feed cuts Ra to a quarter.
A floor
These are kinematic (geometric) values. Built-up edge, chatter, tool wear, the material tearing rather than shearing, and machine vibration all add roughness, so measured Ra is usually higher; a wiper insert flattens the peaks and beats the formula. When the feed exceeds twice the nose radius the arcs no longer overlap and the formula does not apply. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Kinematic roughness of a single-point tool: Rt = r − √(r² − f²/4) ≈ f²/(8r), Ra ≈ f²/(31.2r) for the circular-arc profile; feed for a target Ra f = √(31.2·r·Ra)
Last reviewed 20 September 2026. How results are checked: How we verify.