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CNC Ball Endmill Cusp Height Calculator — Surface Finish Optimizer

Calculate the scallop height left by a ball nose endmill based on tool diameter and step-over distance. Optimize 3D surfacing toolpaths for target Ra values on molds, dies, and freeform surfaces.

CNC Ball Endmill Cusp Height Calculator

Compute the microscopic scallop ridge (cusp) height left on a 3D surface after a ball nose endmill pass. Optimize step-over for your target surface finish.

Max 5 in

Auto-capped at D = 0.5000 in

H = (D/2) − √[(D/2)² − (S/2)²]  |  r = 0.2500, S/2 = 0.0250
Cusp Height (H)
0.001253
in
= 31.8 µm
Est. Ra (Theoretical)
0.000313
in

Ra ≈ H ÷ 4

Roughing
S = 25–40% of D
Semi-Finish
S = 10–20% of D
Finish
S = 3–8% of D

Practical Example

A machinist finishing a mold cavity with a 0.25" ball endmill uses a 0.020" step-over (8% of D) for a fine finish pass. Using H = (0.125) − √(0.125² − 0.010²) = 0.125 − 0.124598 = 0.000402 inches (10.2 µm). This is considered a fine finish achievable without polishing on aluminum. Doubling the step-over to 0.040" quadruples the cusp height to ~40 µm — the non-linear penalty for aggressive stepping.

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What is 3D Surfacing: Scallop Height & Toolpath Optimization?

When a ball nose endmill traverses a surface with a lateral step-over between passes, it leaves behind tiny ridges called cusps or scallops. The height of these ridges determines surface finish: a small cusp height produces a smooth, near-polished surface, while a large cusp height leaves visible grooves requiring hand finishing. The Pythagorean theorem governs the exact geometry.

Mathematical Foundation

Cusp (Scallop) Height
H=D2−(D2)2−(S2)2H = \frac{D}{2} - \sqrt{\left(\frac{D}{2}\right)^2 - \left(\frac{S}{2}\right)^2}
HH= Cusp height — the ridge of uncut material between adjacent passes.
DD= Ball endmill diameter. The tip radius R = D/2.
SS= Step-over distance — lateral offset between consecutive passes. Must be less than D.
Simplified Form
H=R−R2−(S/2)2H = R - \sqrt{R^2 - (S/2)^2}
RR= Ball radius = D/2.
R2−(S/2)2R^2 - (S/2)^2= Pythagorean term — the square of the vertical distance from tool center to the ridge peak.

Laws & Principles

  • Finishing Step-Over Rule: For finish passes, step-over should be 3-8% of tool diameter. At 5% step-over on a 0.500 inch ball mill, cusp height is approximately 0.000625 inch — smooth enough for most molds without polishing.
  • Non-Linear Relationship: Cusp height grows as the square of step-over. Doubling step-over quadruples cusp height. This is why finish passes with small step-overs produce dramatically better surfaces.
  • Tool Diameter Advantage: A larger ball mill produces lower cusp heights at the same step-over percentage but cannot reach tight corners. Choose the largest ball mill that fits the part geometry.
  • Ra Approximation: Theoretical Ra is roughly H/4 for a ball endmill on a flat surface. Actual Ra will be higher due to vibration, runout, and thermal expansion.
  • Scallop vs. Raster Toolpath: Raster (parallel) toolpaths produce constant step-over but varying cusp heights on curved surfaces. Scallop toolpaths maintain constant cusp height across the surface — preferred for optical molds and die work.

Step-by-Step Example Walkthrough

" A mold maker finishes a die cavity with a 0.250 inch ball endmill. The spec requires Ra of 32 micro-inches or less before polishing. "

  1. 1. Target H from Ra: H = 4 x Ra = 4 x 0.000032 = 0.000128 inch.
  2. 2. Solve for S: 0.000128 = 0.125 - sqrt(0.015625 - (S/2)^2).
  3. 3. sqrt(0.015625 - (S/2)^2) = 0.124872.
  4. 4. (S/2)^2 = 0.000032, so S/2 = 0.00566 inch, S = 0.0113 inch (4.5% of D).
  5. 5. Program the finish toolpath with 0.0113 inch step-over.
Final Result: A 0.0113 inch step-over with a 0.250 inch ball mill produces a cusp height of 0.000128 inch (3.25 microns), meeting the Ra target before polishing.

Quick Answer: What Step-Over Gives Me the Surface Finish I Need?

Enter your ball endmill diameter and step-over distance, and this calculator returns the theoretical cusp height in inches and microns, plus the approximate Ra surface roughness. Use it to dial in your finishing toolpath so the scallop height meets print specs without excessive cycle time.

Core Formulas

Cusp Height

H = R - sqrt(R² - (S/2)²)

Where R = ball radius (D/2) and S = step-over distance. Both must be in the same units.

Approximate Surface Roughness

Ra ≈ H / 4

This is a theoretical minimum on flat surfaces. Real-world Ra is higher due to machine vibration, tool runout, and material spring-back.

Solve for Step-Over from Target Cusp Height

S = 2 × sqrt(R² - (R - H)²)

Rearranged form: plug in your target H (from the required Ra × 4) and solve for the maximum step-over.

Real-World Scenarios

✓ Optimized Step-Over Saves 2 Hours of Polishing

A die shop programs a cavity finish pass with a 0.500 inch ball mill at 5% step-over (0.025 inch). Cusp height = 0.500/2 - sqrt(0.0625 - 0.000156) = 0.000625 inch (15.9 microns). Ra ≈ 0.000156 inch (4 microns). The surface goes straight to polishing with minimal stoning, saving 2 hours of bench work compared to a 10% step-over that would have produced 0.0025 inch cusps.

✗ Oversized Step-Over Creates Visible Grooves

A programmer uses 15% step-over (0.075 inch) on the same 0.500 inch ball mill to save cycle time. Cusp height jumps to 0.00564 inch (143 microns) — 9× worse than 5% step-over. The grooves are visible to the naked eye and require 4+ hours of hand stoning before polishing can begin. The 20 minutes saved on cycle time cost the shop 4 hours of skilled labor at $65/hr.

Cusp Height vs. Step-Over Reference

Step-Over (% of D) H (0.250 in ball) H (0.500 in ball) H (1.000 in ball) Ra Approx.
3% 0.000023 in 0.000023 in 0.000011 in ~3-6 µin
5% 0.000063 in 0.000063 in 0.000031 in ~8-16 µin
8% 0.000160 in 0.000160 in 0.000080 in ~20-40 µin
10% 0.000250 in 0.000250 in 0.000125 in ~32-63 µin
15% 0.000564 in 0.000564 in 0.000282 in ~70-140 µin

Note: cusp height as a percentage of D is constant regardless of absolute tool diameter. A larger tool covers more area per pass, so absolute step-over (in inches) is larger while cusp height stays the same.

Pro Tips & Common Mistakes

Do This

  • ✓Use the largest ball mill that fits the geometry. A 0.500 inch ball at 5% step-over covers twice the area per pass as a 0.250 inch ball at the same cusp height, cutting cycle time in half.
  • ✓Use scallop toolpaths for curved surfaces. Raster toolpaths produce constant step-over but varying cusp height on 3D geometry. Scallop toolpaths maintain constant H across the entire surface.
  • ✓Measure actual Ra with a profilometer after test cuts. Theoretical H assumes zero vibration, perfect tool geometry, and rigid fixturing. Real-world Ra is typically 1.5-2x the theoretical value.

Avoid This

  • ✗Don't use a flat endmill for 3D surfacing and expect smooth results. Flat endmills leave stair-step artifacts on curved surfaces. Ball nose endmills are the only geometry that produces smooth scallops on freeform shapes.
  • ✗Don't ignore tool runout. A tool with 0.001 inch runout adds that amount to the theoretical cusp height. At 5% step-over, runout can double the effective scallop height. Use shrink-fit or hydraulic holders for finish passes.
  • ✗Don't cut at the very tip of a ball mill. The effective cutting speed at the tip is zero (0 SFM). Contact below 10% of the radius causes rubbing instead of cutting, producing a poor surface and accelerated wear.

Frequently Asked Questions

What step-over should I use for a finish pass on a mold cavity?

For injection molds requiring SPI B-2 or better finish (Ra 4-8 micro-inches after polishing), use 3-5% step-over with the largest ball mill that fits the cavity. For prototype molds or non-cosmetic surfaces, 8-10% is acceptable. Above 10%, the scallops become visible and require significant bench time to remove before polishing.

Why does doubling the step-over quadruple the cusp height?

The cusp height formula contains S² in the Pythagorean term. For small step-overs relative to tool diameter, H approximates to S² / (8R). Since S is squared, doubling it increases H by a factor of four. This non-linear relationship is why small reductions in step-over produce dramatic improvements in surface finish — and why aggressive step-overs destroy finish quality far faster than most programmers expect.

How do I convert cusp height to Ra surface roughness?

The theoretical relationship is Ra ≈ H/4 for a ball endmill on a flat surface. This assumes the cusp profile is a perfect circular arc. In practice, vibration, tool deflection, and material spring-back increase actual Ra to 1.5-2x the theoretical value. Always run a test cut on scrap material and measure with a profilometer before committing to a finishing strategy on the actual part.

What is the difference between raster and scallop toolpaths for 3D finishing?

Raster (parallel line) toolpaths maintain a constant XY step-over, but on curved surfaces the actual 3D step-over varies — steep slopes get wider spacing and taller cusps. Scallop (constant cusp) toolpaths adjust the XY spacing at every point to maintain a uniform cusp height across the entire surface. Scallop paths produce better surface quality on complex 3D geometry and are standard for optical molds, turbine blades, and medical implant cavities.

Can I use this calculator for tapered ball endmills?

Yes, as long as you use the ball tip diameter (not the shank diameter) for D. A tapered ball endmill with a 0.125 inch tip and 3-degree taper still produces cusps based on the 0.125 inch ball radius at the cutting point. The taper provides rigidity but does not change the scallop geometry. The only exception is if the tapered section itself contacts the workpiece on steep walls — in that case, the effective radius at the contact point is larger and the cusp height will be lower than the tip-only calculation predicts.

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