Surface roughness parameters — Ra, Rz, Rt and beyond

“Ra 1.6” is the most common surface specification on a machining drawing, and one of the most misunderstood. A single number cannot describe a surface; each parameter answers one specific question about the profile. Knowing which question it answers is what makes the number useful.

The profile and the mean line

A stylus instrument traces the surface along a line and records the heights. After the form is removed, a filter with the cut-off wavelength λc separates the short-wavelength roughness from the longer-wavelength waviness. All R-parameters are calculated from this roughness profile, measured from its mean line.

Under ISO 4287 and ISO 4288 the profile is evaluated over sampling lengths lr equal to λc; ISO 4288 sets the default evaluation length at five sampling lengths.

mean line±Ra bandhighest peak and deepest valley in the section

Height parameters

Ra, the arithmetic mean deviation, is the mean of the absolute heights Z(x) from the mean line:

$$Ra = \frac{1}{l}\int_0^{l} |Z(x)|\,dx$$

Rq is the root mean square of the same heights. It weights large deviations more strongly than Ra.

Rp and Rv are the height of the highest peak and the depth of the deepest valley within a sampling length.

Rz, the maximum height of the profile, is Rp + Rv within one sampling length. The value reported is the mean of the five values Rz_i over the five sampling lengths; the largest of them is Rz1max.

$$Rz = \frac{1}{5}\sum_{i=1}^{5} Rz_i \qquad Rz_i = Rp_i + Rv_i$$

Rt, the total height of the profile, is the highest peak plus the deepest valley over the whole evaluation length. It is therefore always at least as large as Rz.

Ra and Rq average everything, so a single scratch or burr barely changes them. Rz responds more to individual peaks and valleys, and Rt is set by the single worst feature. There is no fixed conversion between Ra and Rz: the ratio depends on the shape of the profile. Sunnen, for example, gives Rz ≈ 7.6 · Ra for surfaces honed with a single grit, and states that its factors do not apply to composite-grit finishes.

Shape, spacing and material ratio

• Rsk (skewness) is the mean of Z³ divided by Rq³. It is negative when deep valleys interrupt a flat top surface, and positive when isolated peaks stand above it. Two surfaces with the same Ra can have opposite Rsk.
• Rku (kurtosis) is the mean of Z⁴ divided by Rq⁴. It equals 3 for a Gaussian profile; higher values mean sharper peaks or valleys.
• RSm is the mean width of the profile elements, the spacing of the texture along the surface. In turning it reflects the feed.
• Rmr(c), the material ratio, is the share of the evaluation length in which the profile is cut by a line at height c, in percent. Its curve over all heights, the Abbott–Firestone curve, is the basis of the Rk parameters.

Rsk < 0

Deep valleys interrupt a flat top surface.

Ra 1.00 · Rsk −1.52 · Rku 6.1

Rsk ≈ 0

Heights spread evenly on both sides of the mean line.

Ra 1.00 · Rsk −0.01 · Rku 3.0

Rsk > 0

Isolated peaks stand above the surface.

Ra 1.00 · Rsk 1.52 · Rku 6.1

±Ramean lineheight distribution

Choosing the cut-off (ISO 4288)

The cut-off λc must suit the roughness being measured. ISO 4288:1996 selected it from the expected value of the parameter:

ISO 3274 pairs each cut-off with a short-wave filter λs and a maximum stylus tip radius: 2.5 µm and 2 µm for λc up to 0.8 mm, 8 µm and 5 µm for 2.5 mm, and 25 µm and 10 µm for 8 mm.

Under ISO 4288 a specification such as Ra 1.6 was met when no more than 16 % of the measured values exceeded it (the 16 % rule). “Ra max 1.6” required every value to stay within it.

What changed with ISO 21920

ISO 21920-1, -2 and -3, published on 20 December 2021, replaced ISO 1302 (drawing indication), ISO 4287 (parameters) and ISO 4288 (measurement rules); ISO 21920-2 also replaced ISO 13565-2 and -3.

Replaced standardsISO 21920, 20 December 2021

• ISO 1302drawing indication

ISO 21920-1drawing indication

• ISO 4287parameters
• ISO 13565-2Rk parameters
• ISO 13565-3

ISO 21920-2parameters

• ISO 4288measurement rules

ISO 21920-3measurement rules

The main changes:

The Ra bands for choosing the cut-off stayed the same. The Rz bands moved: an upper limit of Rz up to 0.16, 0.8, 16 or 80 µm now selects N_ic = 0.08, 0.25, 0.8 or 2.5 mm, and above 80 µm, 8 mm.

Typical Ra by process

Process502512.56.33.21.60.80.40.20.10.050.025

Sawing

Planing, shaping

Drilling

Milling

Broaching

Reaming

Boring, turning

Roller burnishing

Grinding

Honing

Polishing

Lapping

Superfinishing

average application less frequent application← rougher · Ra (µm) · finer →

In turning, the feed and the nose radius set the geometric part of the roughness. The theoretical roughness calculator shows how far that sets the limit.