Turned Surface Finish (Ra) Calculator

How it works

Theoretical Rmax (Rz) = f² ÷ (8 r); Ra ≈ f² ÷ (18√3 r) = f² ÷ (31.2 r) (f in mm/rev, r in mm, result in mm → ×1,000 for µm); exact cusp Rt = r − √(r² − f²/4); practical Ra ≈ 1.3–1.8 × theoretical; feed for target Ra: f = √(Ra ÷ factor × 31.2 r); nose radius for target: r = f² ÷ (31.2 × Ra ÷ factor); rpm = Vc × 1,000 ÷ (π D)

A turned surface is a helical groove left by the insert nose. Its theoretical peak-to-valley height is set purely by geometry: the feed squared over eight times the nose radius. For the arc-shaped cusp, Ra works out to the feed squared over 18√3 (about 31.2) times the radius, roughly a quarter of the peak-to-valley height; some shop charts round the divisor to 32, which reads about 2.5% lower. So halving the feed quarters the roughness, and doubling the nose radius halves it. That is why finishing passes use small feeds and large-radius or wiper inserts, whose flat land smears the peaks. Real surfaces are rougher than the formula by 30–80% because of built-up edge, tool wear, vibration and material tearing, so the calculator applies a condition factor that you can replace with your own measured ratio. You can work in either direction: enter a feed to see the finish, or enter the Ra or Rz on the drawing to get the largest feed or the smallest nose radius that will hold it. The drawing default of Ra 3.2 µm is easy at 0.25 mm/rev with a 0.8 mm nose; Ra 0.8 needs about 0.12 mm/rev and a good setup; below 0.4 is grinding territory.

Worked example

Feeding 0.2 mm/rev with a 0.8 mm nose radius: theoretical Ra = 0.2² ÷ (31.18 × 0.8) × 1,000 = 1.60 µm (Rz 6.25 µm), or about Ra 2.08 µm (82 µin, grade N8) in normal production. To hold Ra 1.6 µm reliably the feed should drop to about 0.175 mm/rev, or the nose radius should grow to about 1.04 mm (a 1.2 mm insert) at the same feed. A wiper insert could stay near 0.35 mm/rev. On a 50 mm bar at 200 m/min that is 1,273 rpm and 255 mm/min, so a 50 mm long finishing pass takes about 12 seconds.

Sources: Sandvik Coromant — turning formulas and definitions (surface roughness, rpm, cutting time) · Tribonet — turning surface finish from feed and nose radius (exact and approximate cusp models)

Frequently asked questions