Theoretical roughness Ra / Rz

Feed and nose radius leave a row of cusps on the surface. This tool draws the cusp with its exact geometry, computes Ra and Rt from the profile, and shows the textbook approximation beside it.

Live

Insert / holder presetFeed f 0.20 mm/revNose radius rε mm

Entering angle κr °Minor cutting edge angle κr′ °

—µmRa (geometric)

—µmRz = Rt (geometric)

—µmRa, textbook

—µmRt, textbook

At this feed only the nose radius forms the cusp; the textbook formula is exact.

Ra map for the current angles (µm)

Rows are feed (mm/rev), columns nose radius (mm). Computed live with the κr and κr′ set above.

f ↓ · rε →	0.2	0.4	0.8	1.2	1.6
0.05	—	—	—	—	—
0.1	—	—	—	—	—
0.15	—	—	—	—	—
0.2	—	—	—	—	—
0.25	—	—	—	—	—
0.3	—	—	—	—	—
0.4	—	—	—	—	—
0.5	—	—	—	—	—

Formulae

R_{t} = \frac{f^{2}}{8 r_{ε}}

R_{a} \approx \frac{f^{2}}{31.2 r_{ε}} = \frac{f^{2}}{18 \sqrt{3} r_{ε}}

f and rε in mm; result in mm, ×1000 for µm. Geometric values are computed numerically from the lower envelope of the tool profile repeated every f, including the κr and κr′ edges.

Notes

Values are ideal geometry: vibration, built-up edge, tool wear and tearing all raise the measured Ra. In practice the measured value can be 1.5–3 times the theoretical one.
The textbook formula assumes only the nose radius cuts, which holds when f ≤ 2·rε·sin κr′. With a narrow κr′ (5° for CNMG in a 95° holder) that limit is small and the real cusp is lower.
Rz equals Rt here because the profile is periodic: every sampling length repeats the same peak and valley.
The formula applies when the depth of cut exceeds the nose radius (ap > rε); at shallower cuts the effective angles change.

Sources (4)

Sandvik Coromant — General turning formulas and definitions
G. Boothroyd, W. A. Knight, Fundamentals of Machining and Machine Tools, 3rd ed., CRC Press, 2006 — ideal surface roughness in turning (Rt = f²/8rε, Ra ≈ 0.0321·f²/rε)
M. C. Shaw, Metal Cutting Principles, 2nd ed., Oxford University Press, 2005 — geometry of the turned surface
ISO 21920-2:2021 — Surface texture: Profile — Terms, definitions and surface texture parameters