Gear Calculator

Standards and conventions

• Involute geometry per ISO 21771 with the ISO 53 basic rack: addendum 1.0·m, dedendum 1.25·m, pressure angle 20° (14.5° and 25° also available).
• Helical gears use the normal system: normal module mn and normal pressure angle αn.
• Standard modules per ISO 54; diametral pitch converts with m = 25.4 / DP.
• Worm gears per ISO/DIN 3975 with axial module m, diameter factor q = d1 / m.
• Straight bevel gears: equal addendum, constant clearance, outer (heel) dimensions.

Key formulas

• mt = mn / cos β, αt = atan(tan αn / cos β), d = z·mt, db = d·cos αt
• da = d + 2·mn·(1 + x), df = d − 2·mn·(1.25 − x)
• inv α = tan α − α; the inverse is solved by Newton–Raphson to 1e-12.
• Span: Wk = mn·cos αn·[π(k − 0.5) + z·inv αt] + 2x·mn·sin αn
• Pair: inv αwt = 2·tan αn·(x1 + x2)/(z1 + z2) + inv αt, a = a0·cos αt / cos αwt
• Bevel: δ1 = atan(sin Σ / (z2/z1 + cos Σ)), Re = d1 / (2·sin δ1)
• Worm: γ = atan(z1 / q), η = tan γ / tan(γ + ρ′)

Frequently asked questions

How do I find the module of an unknown gear?

Count the teeth z and measure the tip diameter da. For a spur gear without profile shift, m = da / (z + 2). Also measure the span Wk over k teeth: with both measurements the calculator solves the module and the profile shift together, then snaps to the nearest ISO 54 module or diametral pitch.

Is my gear metric or imperial?

Compare the estimated module with the nearest ISO 54 module and the nearest diametral pitch converted with m = 25.4 / DP. The smaller deviation usually tells you the system; for example DP 8 is module 3.175 mm, close to but not the same as module 3.

Why is span measurement more reliable than tip diameter?

Tip diameters are often reduced by tip chamfer, tip shortening or wear. The span (base tangent length) is measured across tooth flanks at the base circle tangent, so it is not affected by the tip and directly reflects the tooth thickness and profile shift. Measure over unworn teeth.

What contact ratio is acceptable for a spur gear pair?

The transverse contact ratio εα should be at least 1.2 and preferably 1.4 or more for smooth running. Helical gears add the overlap ratio εβ = b·sin β / (π·mn).

Does this calculator check gear strength?

No. It covers geometry only, per ISO 21771 with the ISO 53 basic rack. Load capacity to ISO 6336 or AGMA 2001 is not covered; send the specification to Misuba engineering for critical parts.