Helical Gear Geometry Calculator
Normal module, helix angle & teeth count → full helical geometry — ISO 54 standard
Normal Module (mₙ) — ISO standard
Helix Angle (β)
deg
Normal Pressure Angle (αₙ)
Pinion Teeth (z₁)
teeth
Gear Teeth (z₂)
teeth
Face Width (b)
mm
Input Speed (n₁)
RPM
Typical Helix Angles
Applicationβ
Automotive15°–30°
Industrial gearboxes10°–25°
High-speed25°–45°
Low noise20°–35°
Higher β = smoother operation, more axial thrust. Keep β < 45° to limit axial load.
Transverse Plane Parameters
Transverse Module (mₜ)
—mm
Transverse Pressure Angle (αₜ)
—deg
Transverse Circular Pitch (pₜ)
—mm
Normal Circular Pitch (pₙ)
—mm
Axial Pitch (pₓ)
—mm
Center Distance (a)
—mm
Pinion Dimensions (Gear 1)
Pitch Circle Diameter (d₁)
—mm
Tip Diameter (dₐ₁)
—mm
Root Diameter (dₑ₁)
—mm
Base Circle (d⁢₁)
—mm
Lead (L₁)
—mm
Virtual Teeth (zᵥ₁)
—teeth
Gear Dimensions (Gear 2)
Pitch Circle Diameter (d₂)
—mm
Tip Diameter (dₐ₂)
—mm
Root Diameter (dₑ₂)
—mm
Base Circle (d⁢₂)
—mm
Lead (L₂)
—mm
Virtual Teeth (zᵥ₂)
—teeth
Contact Ratios & Speed
Gear Ratio (i)
—
Output Speed (n₂)
—RPM
Transverse Contact Ratio (εα)
—
Axial (Overlap) Ratio (εβ)
—
Total Contact Ratio (εγ)
—
Pitch Line Velocity
—m/s
Key Formulas
Transverse modulemₜ = mₙ / cos(β)
Transv. pressure angletan(αₜ) = tan(αₙ) / cos(β)
Pitch circle dia.d = mₜ × z
Tip diameterdₐ = d + 2mₙ
Root diameterdₑ = d − 2.5mₙ
LeadL = π × d / tan(β)
Axial pitchpₓ = π × mₜ / tan(β)
Virtual teethzᵥ = z / cos³(β)
Overlap ratioεβ = b × sin(β) / (π × mₙ)