Enter the tool radius (for a ball-nose tool) and the stepover into the calculator to determine the scallop height (also called cusp height) for a milling operation. Scallop height is a geometric measure of the residual ridges left between adjacent toolpaths and is commonly used to estimate theoretical surface finish in 3D machining.
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Scallop Height Formula
For a ball-nose tool, scallop height (also called cusp height) is the theoretical ridge left between two adjacent toolpaths. It is commonly used in CNC finishing to estimate how smooth the surface will be for a given tool radius and stepover.
h = R - \sqrt{R^2 - \left(\frac{s}{2}\right)^2}- h = scallop height
- R = ball-nose tool radius
- s = stepover between adjacent passes
If your cutter is specified by diameter instead of radius, convert it first:
R = \frac{D}{2}Use the same unit for every input. If radius and stepover are entered in millimeters, the scallop height will also be in millimeters. The same rule applies for inches.
Rearranged Forms
This calculator can solve for any missing variable, so the equivalent forms below are useful when you want to size a stepover from a target finish or back-calculate the required tool radius.
Solve for stepover:
s = 2\sqrt{2Rh - h^2}Solve for tool radius:
R = \frac{s^2 + 4h^2}{8h}Valid Input Range
In this ball-nose geometry model, the stepover cannot exceed the tool diameter. A larger value would make the square-root term invalid and no longer represent adjacent passes of the same spherical tip.
0 \leq s \leq 2R
Practical interpretation:
- Smaller stepover → smaller scallop height
- Larger radius → smaller scallop height for the same stepover
- Larger stepover → taller cusps and a rougher theoretical finish
How to Calculate Scallop Height
- Identify the ball radius of the cutter. If you only know the diameter, divide it by 2.
- Measure or choose the stepover between adjacent passes.
- Keep both values in the same unit system.
- Substitute into the formula to compute the scallop height.
- If you are planning a finish pass, compare the result to your allowable cusp target and reduce the stepover if needed.
Small-Step Approximation
When the stepover is small compared with the tool radius, the exact formula simplifies to a very useful approximation:
h \approx \frac{s^2}{8R}If you prefer diameter form, substitute R = D/2:
h \approx \frac{s^2}{4D}This approximation shows an important machining rule: scallop height changes approximately with the square of stepover. In other words, if you halve the stepover, the scallop height drops to about one-quarter. If you double the stepover, the scallop height becomes about four times larger.
Example Calculations
Example 1: Find Scallop Height from Radius and Stepover
A 0.50 in diameter ball-nose tool has a radius of 0.25 in. If the stepover is 0.02 in, then:
h = 0.25 - \sqrt{0.25^2 - \left(\frac{0.02}{2}\right)^2} \approx 0.000200 \text{ in}That is approximately 0.00508 mm. This is a very small cusp, which is why small finishing stepovers can produce a much smoother theoretical surface.
Example 2: Find Maximum Stepover from a Target Cusp Height
If the tool radius is 5 mm and the desired scallop height is 0.01 mm, solve for the largest allowable stepover:
s = 2\sqrt{2(5)(0.01) - 0.01^2} \approx 0.632 \text{ mm}Any stepover larger than about 0.632 mm would increase the theoretical cusp above the 0.01 mm target.
What Affects Scallop Height?
| Factor | Effect on Scallop Height | Why It Matters |
|---|---|---|
| Smaller stepover | Decreases scallop height quickly | The most direct way to improve theoretical finish quality |
| Larger ball radius | Decreases scallop height at the same stepover | Larger tools can often use wider passes for the same cusp target |
| Smaller ball radius | Increases scallop height at the same stepover | Small tools usually need tighter stepovers to maintain finish |
| Inconsistent units | Produces incorrect results | Radius, stepover, and scallop height must all use the same unit basis |
| Tighter finish target | Requires smaller stepover | Improves surface quality but usually increases cycle time |
Interpreting the Result
The value from this calculator is a geometric estimate. It is best used as a planning tool for toolpath spacing and finish-pass setup.
- Scallop height is not the same as measured surface roughness. It describes the ideal cusp left by tool geometry, not every real machining effect.
- Actual finish can differ because of tool wear, runout, machine accuracy, material behavior, deflection, and CAM strategy.
- This model is for a ball-nose tool. Other cutter shapes use different geometry.
- For finish machining, choose stepover from the target scallop height rather than guessing a percentage of tool diameter.
Common Usage Notes
- If you know the desired finish quality, start with a target h and solve for s.
- If you already have a programmed stepover, calculate h to estimate how visible the cusp may be.
- If cycle time is too high, increasing tool radius can reduce scallop height without forcing as small a stepover.
- For very small stepovers, the approximation formula is often fast and accurate enough for quick planning.
