August 2026 13 min read

Helical Gear Calculator

Free helical gear calculator. Compute transverse module, pitch diameter, axial pitch, lead, minimum face width and transverse pressure angle from normal module, teeth and helix angle, plus the axial thrust factor.

Helical Gear Calculator

Helical Gear Inputs

Unit System

SI — Normal Module (mm) Imperial — Normal DP

Normal Module m_n (mm)

Tooth size normal to face

Gear 1 Teeth z₁

Gear 2 Teeth z₂

Helix Angle ψ (°)

8–30° typical

Normal Pressure Angle

Auto-calculate instantly

Center Distance

82.82mm

d₁ = 41.4 mm, d₂ = 124.2 mm.

Thrust is moderate; standard bearings usually suffice

Transverse Module

2.071mm

Pitch Dia d₁

41.41mm

Pitch Dia d₂

124.23mm

Axial Pitch

24.28mm

Lead

485.5mm

Min Face Width

27.92mm

Transv. Press. Angle

20.65°

Thrust Factor

0.2679

Transverse Module

2.071mm

m_n/cos(ψ)

Pitch Dia d₁

41.41mm

m_t·z₁

Pitch Dia d₂

124.23mm

m_t·z₂

Axial Pitch

24.28mm

p_n/sin(ψ)

Lead

485.5mm

π·d/tan(ψ)

Min Face Width

27.92mm

1.15·p_x

Transv. Press. Angle

20.65°

tan(α_t)=tan(α_n)/cos(ψ)

Thrust Factor

0.2679

tan(ψ)

Result Summary

Normal Module	2 mm
Helix Angle	15°
z₁ / z₂	20 / 60 T
Pitch d₁	41.41 mm
Pitch d₂	124.23 mm
Center Distance	82.82 mm
Axial Pitch	24.28 mm
Min Face Width	27.92 mm
Thrust Factor	0.2679

Estimated Result — Uses standard helical geometry: m_t = m_n/cos(ψ), d = m_t·z, p_x = p_n/sin(ψ), L = π·d/tan(ψ), b_min = 1.15·p_x, and tan(α_t) = tan(α_n)/cos(ψ). The thrust factor tan(ψ) is the axial force per unit tangential force. Profile shift, contact-ratio detail and AGMA/ISO strength rating are not included.

Interactive Diagram

Helical teeth are angled by ψ; the transverse module grows as m_n/cos(ψ).

Visualizations

Helical Gear Dimensions (mm)

41mmd₁

124mmd₂

24.3mmp_x

486mmL

27.9mmb_min

83mmC

Axial Thrust Factor

tan(ψ)27%

Formula

Helical gears are defined by the normal module (the design tooth size) and the helix angle. The transverse module — used for pitch diameter — is larger by 1/cos(ψ). Axial pitch and lead follow from the helix, and the minimum face width ensures continuous tooth engagement. Axial thrust scales with tan(ψ).

m_t = m_n / cos(ψ)

Alternative Form

d = m_t·z , p_x = p_n/sin(ψ) , L = π·d/tan(ψ) , b_min = 1.15·p_x

Variables

Symbol	Name	Unit	Description
m_n	Normal module	mm	Tooth size normal to face
m_t	Transverse module	mm	m_n / cos(ψ)
ψ	Helix angle	°	Tooth trace angle to axis
p_x	Axial pitch	mm	p_n / sin(ψ)
L	Lead	mm	π·d / tan(ψ)
b_min	Min face width	mm	1.15·p_x
tan(ψ)	Thrust factor	—	Axial force / tangential force

Explanation

The helix angle is the key extra degree of freedom versus a spur gear. It stretches the transverse module (d = m_n·z/cos(ψ)), so a helical gear is slightly larger than a spur of the same normal module and teeth. In return it gains overlap contact (smoother, quieter) but produces axial thrust F_a = F_t·tan(ψ) that the bearings must carry. The minimum face width of about 1.15·p_x guarantees at least one full tooth is always in mesh.

Derivation

Normal and transverse pitches relate by p_t = p_n/cos(ψ), so m_t = m_n/cos(ψ) and d = m_t·z. Axial pitch p_x = p_n/sin(ψ). Lead L is the axial advance per turn: L = π·d/tan(ψ). Minimum face width b_min ≈ 1.15·p_x keeps continuous engagement. Transverse pressure angle follows tan(α_t) = tan(α_n)/cos(ψ) from the tooth geometry projection.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Find the transverse module from the normal module.

m_t = m_n / cos(ψ)

m_n:2 mm

ψ:15°

Intermediate: cos(15°) = 0.9659

Result: m_t = 2.071 mm

Step 2

Compute the pitch diameters.

d = m_t·z

z₁:20

z₂:60

Intermediate: d₁ = 2.071×20, d₂ = 2.071×60

Result: d₁ = 41.41 mm, d₂ = 124.23 mm

Step 3

Compute axial pitch and lead.

p_x = p_n/sin(ψ) , L = π·d/tan(ψ)

p_n:6.283 mm

Intermediate: p_x = 6.283/sin(15°)

Result: p_x = 24.28 mm, L = 485.5 mm

Step 4

Set minimum face width (1.15·p_x).

b_min = 1.15·p_x

p_x:24.28 mm

Intermediate: b_min = 1.15 × 24.28

Result: 27.92 mm

Step 5

Center distance and transverse pressure angle.

C = (d₁+d₂)/2 , tan(α_t) = tan(α_n)/cos(ψ)

α_n:20°

Intermediate: C = (41.4+124.2)/2

Result: C = 82.82 mm, α_t = 20.65°

Final Answer

82.82 mm

Worked Examples

See how the calculator works with different scenarios.

Scenario

m_n = 2 mm, 20/60 teeth, ψ = 15°.

Inputs

m_n:2 mm

z₁/z₂:20/60

ψ:15°

Calculation

m_t = 2/cos15° = 2.071; d₁ = 41.4, d₂ = 124.3 mm; p_x = π·2/sin15° = 24.25; b_min = 27.9; C = 82.8 mm; thrust = tan15° = 0.268

Result

C = 82.8 mm, b_min = 27.9 mm

Interpretation

A modest helix angle keeps thrust low while smoothing engagement.

Understanding Your Result

Your Result: 82.82 mm

Typical range: Helix angles of 15–25° are common. Your ψ = 15° gives a thrust factor of 0.2679 (moderate).

What This Means

With m_n = 2 mm, z₁ = 20, z₂ = 60, ψ = 15°, the pitch diameters are 41.4 and 124.2 mm, center distance 82.82 mm. Minimum face width 27.92 mm; axial thrust factor 0.2679.

Why It Matters

Helical geometry sets how quietly and strongly the gear runs and how much thrust the bearings must take. Choosing ψ trades smoothness against thrust; getting face width right keeps tooth engagement continuous. These dimensions feed the AGMA/ISO strength rating.

Professional Interpretation

Select thrust bearings sized for F_t·0.2679; keep face width ≥ 27.92 mm; verify bending/contact per AGMA 2001/ISO 6336, accounting for overlap ratio ε_β = b·sin(ψ)/(π·m_n).

Size vs spur

d = m_n·z/cos(15°) is larger than the spur equivalent.

Thrust

F_a = F_t × 0.2679 (tan ψ).

Face width

b_min = 27.92 mm (1.15·p_x).

Recommendations

•Thrust is moderate; standard bearings usually suffice.
•Keep face width ≥ 1.15·p_x for continuous engagement.
•Use 15–25° helix for a good smoothness/thrust trade-off.
•Confirm opposite hand on parallel meshing gears.
•Rate tooth bending and pitting per AGMA/ISO afterwards.

Educational Content

Introduction

Helical gears are the quiet, strong cousins of spur gears: their teeth are cut at a helix angle so engagement is gradual instead of sudden. That single angle changes the geometry — and introduces axial thrust the bearings must handle. This calculator turns the normal module, tooth counts and helix angle into every key helical dimension, including the minimum face width and thrust factor.

Concept Explanation

The design starts from the normal module m_n (tooth size perpendicular to the face). The transverse module m_t = m_n/cos(ψ) is what sets the pitch diameter d = m_t·z, so a helical gear is slightly larger than a spur of the same m_n and teeth. From the helix come the axial pitch p_x = p_n/sin(ψ) and lead L = π·d/tan(ψ); the rule b_min ≈ 1.15·p_x keeps at least one full tooth engaged. Axial thrust scales as F_a = F_t·tan(ψ).

Applications

▸Sizing helical pairs from normal module and ratio

▸Choosing a helix angle for noise vs thrust

▸Setting minimum face width for continuous mesh

▸Estimating axial thrust for bearing selection

▸Teaching transverse vs normal geometry

▸Feeding AGMA/ISO helical strength ratings

Benefits

•Returns full helical geometry from m_n, z and ψ
•Computes axial pitch, lead and min face width
•Gives transverse pressure angle and thrust factor
•SI / imperial consistent
•Diagrams the helix geometry

Limitations

•No profile shift
•No contact-ratio detail
•No AGMA/ISO strength rating
•Parallel-shaft pair assumed
•Thrust is a factor, not a force (needs F_t)

Best Practices

•Pick m_n from a standard series
•Use 15–25° helix for balance
•Keep face width ≥ 1.15·p_x
•Size thrust bearings for F_t·tan(ψ)
•Rate per AGMA 2001/ISO 6336

Common Misconceptions

Myth: Helical and spur use the same module.

Truth: d uses transverse m_t = m_n/cos(ψ), larger than m_n.

Myth: Helix angle only affects noise.

Truth: It also sets thrust, size and minimum face width.

Myth: More helix is always better.

Truth: It raises axial thrust and bearing load.

Myth: Helical needs no thrust bearing at low ψ.

Truth: Any ψ>0 produces thrust; size bearings accordingly.

Practical Tips

•Center distance 82.82 mm; face width ≥ 27.92 mm.
•Thrust is moderate; standard bearings usually suffice.
•Keep face width ≥ 1.15·p_x for continuous engagement.
•Parallel helical gears must have opposite hand.
•Follow with AGMA/ISO strength rating for the load.

Common Mistakes

Using m_n directly for d

Use transverse m_t = m_n/cos(ψ) for pitch diameter.

Ignoring thrust

Size thrust bearings for F_t·tan(ψ); herringbone cancels it.

Face width too small

Keep b ≥ 1.15·p_x so teeth stay engaged.

Mismatched hand

Parallel helical meshes need opposite-hand gears.

Skipping strength check

Run AGMA/ISO bending and pitting ratings.

Assumptions

This calculator makes the following assumptions to provide estimates:

Standard helical tooth: a = m_n, b = 1.25 m_n.

Valid helix: 0 < ψ < 90°.

No profile shift: Standard (unshifted) tooth.

Parallel shafts: Opposite-hand meshing pair.

No strength rating: AGMA/ISO capacity is separate.

Limitations

Be aware of these limitations when using this calculator:

No shift/contact ratio: Overlap ratio not carried into strength.

No strength: Bending/pitting needs AGMA/ISO 6336.

Thrust factor only: Actual thrust needs tangential force F_t.

Single pair: One mating pair, parallel shafts.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a helical gear?

A helical gear has teeth cut at a helix angle to the axis. Gradual engagement makes it smoother and quieter than a spur gear, and it can drive parallel or crossed (non-parallel) shafts.

What is the helix angle?

ψ is the angle between the tooth trace and the gear axis. Common 8–30°; smaller angles cut axial thrust, larger angles improve smoothness and capacity.

What is normal module vs transverse module?

Normal module m_n is measured perpendicular to the tooth face; transverse module m_t is in the rotation plane. m_t = m_n / cos(ψ), and pitch diameter uses m_t: d = m_t·z.

How is helical pitch diameter calculated?

d = m_t·z = m_n·z / cos(ψ). The transverse module is always larger than the normal module.

What is axial pitch?

p_x is the distance between adjacent teeth measured along the axis: p_x = p_n / sin(ψ) = π·m_n / sin(ψ). It sets the minimum face width.

What is lead?

L is the axial travel of a tooth in one revolution: L = π·d / tan(ψ) = π·m_n·z / sin(ψ). Important for hob setup and thrust direction.

What is the minimum face width?

b_min ≥ p_x; a rule of thumb uses b ≥ 1.15·p_x so at least one full tooth is always engaged for smooth running.

What are the advantages over spur gears?

Smoother/quieter, higher load capacity, higher contact ratio (1.5–3.0), less vibration, and ability to drive non-parallel shafts.

What are the disadvantages?

Axial thrust needs thrust bearings, efficiency is slightly lower (96–98%), manufacturing is more complex, and bearing loads rise with helix angle.

How is axial thrust calculated?

F_a = F_t·tan(ψ), where F_t is the tangential force. Smaller ψ reduces thrust but also the smoothness benefit; thrust bearings must handle it.

What is a double helical (herringbone) gear?

Two oppositely-handed helixes on one blank cancel the axial thrust, removing the thrust-bearing need. Used in marine drives and heavy gearboxes.

What is the normal pressure angle?

α_n is the pressure angle in the plane normal to the tooth. Standard 20° (or 25°). The transverse angle α_t satisfies tan(α_n) = tan(α_t)·cos(ψ).

What is contact ratio in helical gears?

Higher than spur: typically 1.5–3.0. Total = transverse contact ratio + face (overlap) ratio. Higher is smoother and quieter.

Can helical gears be used on non-parallel shafts?

Yes — crossed helical (screw) gears at, e.g., 90°, but they have point contact and lower capacity. For high power, use bevel or worm gears.

What is the efficiency of helical gears?

Typically 96–98% per mesh for parallel shafts; crossed helical gears drop to 70–90% due to sliding.

What materials are used?

Case-carburized steel (8620, 4320) for highest loads, through-hardened (4140, 4340) medium, nitrided for low distortion, plastic for light duty.

How are helical gears manufactured?

Usually hobbed with the hob set at the helix angle; also shaping, grinding (precision) and skiving (internal). The angle needs special setup.

What is the minimum teeth for helical gears?

z_min = 17·cos³(ψ) at 20° normal pressure angle. At 15° helix: 17·0.901 ≈ 15.3 teeth — fewer than the spur minimum.

What is the center distance of helical gears?

C = (d₁+d₂)/2 = m_t(z₁+z₂)/2 = m_n(z₁+z₂)/(2·cos(ψ)). The helix angle enters through m_t.

How does helix angle affect strength?

A larger ψ increases effective face contact, lowers load per tooth length and improves distribution, raising capacity — but also raises thrust.

What is helical overlap ratio?

ε_β = b·sin(ψ)/(π·m_n), the extra contact from the helix. Total contact ratio = transverse ratio + ε_β.

How does helix angle affect noise?

Larger ψ gives gradual engagement and lower noise; herringbone is quieter still. Helical beats spur clearly at speed.

What is the thrust load direction?

Depends on hand (right/left) and rotation. Right-hand with clockwise gives one direction; reversing rotation reverses thrust.

How do I select helix angle?

Trade noise (higher better) against thrust (lower better) and manufacturing. 15–25° is a common industrial compromise.

What are standard helix angles?

8°, 10°, 12°, 15°, 20°, 25°, 30°; 15° and 20° most common, 25–30° in automotive transmissions.

How does helix angle affect bearing selection?

Higher ψ means more axial thrust, needing angular-contact or tapered roller bearings oriented for the thrust direction.

Can helical and spur gears be meshed?

No — different profiles; all gears in a mesh must share the same helix angle (opposite hand for parallel shafts).

What is the transverse pressure angle?

α_t in the rotation plane, larger than α_n: tan(α_t) = tan(α_n)/cos(ψ). Used in transverse-plane calculations.

How do I calculate pitch-line velocity?

v = π·d·n/60000 m/s with d in mm, n in rpm — same form as spur using the helical pitch diameter.

What is the AGMA standard for helical gears?

AGMA 2001-D04 rates tooth bending and surface durability, accounting for helix angle via the overlap ratio.

How do I measure helix angle?

With a gear measuring machine, lead checker or CMM; or from ψ = arctan(π·d/L) using a measured lead L.

What is the difference between single and double helical?

Single helical produces thrust; double (herringbone) cancels it but is larger and costlier.

How does helix angle affect gear size?

For the same m_n and z, a helical gear is larger than a spur: d = m_n·z/cos(ψ), growing with ψ.

What lubrication do helical gears need?

An EHL film with good film strength; EP additives are common for high load because of the sliding component.

How do I design a helical gear pair?

Fix power/speed/ratio, choose m_n and ψ, set teeth from ratio and center distance, compute diameters, check face width, verify stress, select thrust bearings.

Why is the transverse module larger than the normal module?

Because m_t = m_n/cos(ψ) and cos(ψ) < 1; the rotation-plane tooth spacing is stretched by the helix.

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Related Guides

Deepen your understanding with these educational resources.

AGMA — Helical gear standards

Helical gear design and rating.

KHK — Helical gear guide

Helical gear technical reference.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Normal Module (m_n): Tooth size perpendicular to the face; the design module.
Transverse Module (m_t): m_t = m_n / cos(ψ); used for pitch diameter.
Helix Angle (ψ): Angle of the tooth trace to the axis.
Axial Pitch (p_x): p_n / sin(ψ); sets minimum face width.
Lead (L): Axial travel per revolution; π·d/tan(ψ).
Overlap Ratio (ε_β): b·sin(ψ)/(π·m_n); extra helical contact.
Thrust Factor: tan(ψ); axial thrust per unit tangential force.
Herringbone: Double-helical gear; thrust cancels.

Authoritative References

Trusted sources and standards referenced by this calculator.

AGMA — Helical gear standards ISO 6336 — Helical gear rating MIT OpenCourseWare — Gear design KHK Gears — Helical gear guide Shigley's Mechanical Engineering Design

Applicable Standards

AGMA 2001-D04ISO 6336ISO 1328DIN 3975

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Last Updated:

August 2026

Version:

2.1.0

References:

License:

Free to use for educational and professional purposes.