AtlasCalcsUtility Calculation Suite
August 2026 13 min read

Worm Gear Calculator

Free worm gear calculator. Compute ratio, center distance, lead angle, efficiency and self-locking of a worm and wheel, with output speed and torque. Handles single to quadruple starts, diameter quotient q and friction.

Worm Gear Calculator

Worm Gear Inputs

Unit System
SI — Module (mm) Imperial — DP
Module m (mm)

Axial module

Worm Starts z₁
Wheel Teeth z₂
Diameter Quotient q

d₁ = q·m

Friction Coefficient μ

Steel/bronze ~0.05

Input Speed (rpm)

Optional

Input Torque (N·m)

Optional

Torque Unit
N·m lbf·in
Auto-calculate instantly

Gear Ratio

40 : 1

Lead angle γ = 5.7°; not self-locking.

Not self-locking: the wheel can back-drive the worm.
Center Dist a

100mm

Worm d₁

40mm

Wheel d₂

160mm

Lead L

12.57mm

Lead Angle γ

5.71°

Efficiency η

66.3%

Self-locking

No

P.L. Velocity

3.14m/s

Center Dist a

100mm

m·(q+z₂)/2

Worm d₁

40mm

q·m

Wheel d₂

160mm

m·z₂

Lead L

12.57mm

π·m·z₁

Lead Angle γ

5.71°

atan(z₁/q)

Efficiency η

66.3%

tanγ/tan(γ+φ)

Self-locking

No

γ ≤ φ

P.L. Velocity

3.14m/s

Worm pitch line

Result Summary

Module4 mm
Worm Starts z₁1
Wheel Teeth z₂40
Ratio40 : 1
q (d₁ = q·m)10
Center Distance100 mm
d₁ / d₂40 / 160 mm
Lead Angle γ5.71°
Friction μ0.050
Efficiency66.3 %
Self-lockingNo
Output Speed37.50 rpm
Output Torque265.33 N·m
Estimated Result — Uses standard worm geometry: d₁ = q·m, d₂ = m·z₂, a = m(q+z₂)/2, L = π·m·z₁, γ = atan(z₁/q), efficiency η = tanγ/tan(γ+φ) with φ = atan μ, self-locking when γ ≤ φ, and output T₂ = T₁·(z₂/z₁)·η. It does not rate bending/wear (AGMA 6022/6034, ISO 14521) nor set heat/cooling or materials.

Interactive Diagram

Lead angle γ sets efficiency and self-locking; the wheel mesh is offset 90° to the worm axis.

Visualizations

Worm Gear Dimensions (mm)

40mmd₁
160mmd₂
100mma
12.6mmL

Efficiency

η66%

Formula

A worm (screw, z₁ starts) meshes with a wheel (z₂ teeth) at 90° offset axes. The ratio is simply z₂/z₁. The worm pitch diameter is d₁ = q·m, the wheel d₂ = m·z₂, center distance a = m(q+z₂)/2, lead L = π·m·z₁ and lead angle γ = atan(z₁/q). Efficiency follows from the lead angle and friction angle, and self-locking occurs when γ ≤ φ.

i = z₂ / z₁

Alternative Form

η = tanγ / tan(γ + φ) , γ = atan(z₁/q) , φ = atan μ

Variables

SymbolNameUnitDescription
mModulemmAxial module (worm and wheel)
z₁Worm starts—1–4 thread count
z₂Wheel teeth—Wheel tooth count
qDiameter quotient—d₁ = q·m
μFriction coeff—Steel/bronze, ~0.05
γLead angle°atan(z₁/q)
φFriction angle°atan μ
ηEfficiency%tanγ/tan(γ+φ)

Explanation

Worm gears pack a big ratio into one stage because the wheel only advances one tooth per worm revolution of a single start. The lead angle is the key: small γ (few starts, large q) gives low efficiency and can self-lock; large γ (more starts, small q) gives high efficiency but no locking. Output torque is the input torque times ratio times efficiency, so a 40:1 worm at 66% still delivers ~26× the input torque.

Derivation

d₁ = q·m, d₂ = m·z₂, a = (d₁+d₂)/2. Lead L = π·m·z₁ (the screw advances L per turn). The helix angle tanγ = L/(π·d₁) = z₁/q, so γ = atan(z₁/q). With friction angle φ = atanμ, the worm-driving efficiency is η = tanγ/tan(γ+φ); the wheel driving (back-drive) efficiency is η_b = tan(γ−φ)/tanγ when γ>φ, else zero (self-locking). Output speed n₂ = n₁/i; output torque T₂ = T₁·i·η.

Step-by-Step Calculation

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

Step 1

Worm and wheel pitch diameters.

d₁ = q·m , d₂ = m·z₂
q:10
m:4 mm
z₂:40
Intermediate: d₁ = 10×4, d₂ = 4×40
Result: d₁ = 40 mm, d₂ = 160 mm
Step 2

Center distance.

a = (d₁ + d₂)/2 = m·(q + z₂)/2
d₁:40 mm
d₂:160 mm
Intermediate: a = (40 + 160)/2
Result: 100 mm
Step 3

Lead and lead angle.

L = π·m·z₁ , γ = atan(z₁/q)
z₁:1
q:10
Intermediate: L = π·4×1; γ = atan(1/10)
Result: L = 12.57 mm, γ = 5.71°
Step 4

Efficiency and self-locking.

η = tanγ / tan(γ + φ) , φ = atan μ
γ:5.71°
μ:0.050
Intermediate: φ = atan(0.05) = 2.86°; γ > φ
Result: η = 66.3%, not self-locking
Step 5

Output speed and torque.

n₂ = n₁/i , T₂ = T₁·i·η
i:40
Intermediate: T₂ = T₁·40·0.6633
Result: 37.50 rpm, 265.33 N·m
Final Answer
40 : 1

Worked Examples

See how the calculator works with different scenarios.

Scenario

m = 4 mm, single-start, 40 teeth, q = 10, μ = 0.05.

Inputs

m:4 mm
z₁/z₂:1/40
q:10
μ:0.05

Calculation

d₁ = 40, d₂ = 160 mm; a = 100 mm; L = π·4·1 = 12.57 mm; γ = atan(1/10) = 5.71°; φ = atan(0.05)=2.86°; η = tan5.71/tan8.57 = 0.66; γ>φ → not locking

Result

40:1, η ≈ 66%

Interpretation

A typical single-stage 40:1 worm; moderate efficiency, not self-locking.

Understanding Your Result

Your Result: 40 : 1

Typical range: Lead angles run ~3–30°. γ = 5.7° → efficient / back-drivable. Efficiency 66% is typical.

What This Means

This worm set has ratio 40:1, center distance 100 mm, lead angle γ = 5.71°, efficiency η = 66.3%, and is not self-locking.

Why It Matters

Worm gears give compact high reductions and optional self-locking, but efficiency collapses as the lead angle drops. Knowing η and the locking condition lets you size the motor, pick the bronze wheel and decide whether a brake is needed.

Professional Interpretation

Rate bending and wear per AGMA 6022/6034 (ISO 14521) using m, z₂, q and the bronze wheel. Allow for heat: at 66% efficiency, about 33.666666666666664% of input becomes heat. Output torque ≈ 265.3 N·m.

Lead angle vs locking

γ = 5.71°, φ = 2.86° → γ > φ (back-drivable).

Heat fraction

33.666666666666664% of input power becomes heat.

Center distance

100 mm.

Recommendations

  • •Not self-locking: add a brake if the load must hold.
  • •Efficiency is acceptable for most drives.
  • •Use a bronze wheel with a hardened steel worm.
  • •Rate capacity per AGMA/ISO; this gives geometry only.
  • •Match q to shaft stiffness (avoid too-small q).

Educational Content

Introduction

A worm gear turns a screw into a huge reduction. This calculator returns the ratio, the worm and wheel diameters, center distance, lead and lead angle, then the efficiency and whether the set self-locks, plus the output speed and torque from your input.

Concept Explanation

Ratio i = z₂/z₁. The worm pitch diameter is d₁ = q·m, the wheel d₂ = m·z₂, so center distance a = m(q+z₂)/2. Lead L = π·m·z₁ is the axial advance per worm turn, and the lead angle γ = atan(z₁/q) sets the performance. Efficiency η = tanγ/tan(γ+φ) with φ = atanμ; if γ ≤ φ the set is self-locking.

Applications

▸Conveyors and material handling
▸Lifts, hoists and winches
▸Valve and tuning actuators
▸Presses and indexing tables
▸Right-angle reductions 5:1 to 100:1
▸Self-locking holding drives

Benefits

  • •Large single-stage ratio
  • •Optional self-locking (no brake)
  • •Quiet right-angle drive
  • •Compact for the reduction
  • •Computes efficiency and locking

Limitations

  • •Lower efficiency than parallel gears
  • •More heat (needs lubrication/cooling)
  • •Bronze wheel increases cost
  • •No bending/wear rating
  • •Self-locking units waste power as heat

Best Practices

  • •Pick starts/q to balance η and locking
  • •Use hardened steel worm + bronze wheel
  • •Rate per AGMA/ISO; allow for heat
  • •Forbid back-driving loads if locking needed
  • •Align axes at 90°, set center distance precisely

Common Misconceptions

Myth: Worm gears are always self-locking.
Truth: Only when lead angle ≤ friction angle (small γ).
Myth: High ratio means high efficiency.
Truth: Small γ (high ratio, single-start) is often low efficiency.
Myth: Efficiency close to 100%.
Truth: Single-start worms can be 30–50% efficient.
Myth: Any bronze works.
Truth: Match bronze grade to load/speed (AGMA).

Practical Tips

  • •Ratio 40:1 at 66% efficiency.
  • •Not self-locking: consider a brake for holding.
  • •More starts → larger lead angle → higher η, less locking.
  • •Use a bronze wheel with a hardened steel worm.
  • •Plan cooling when efficiency is low.

Common Mistakes

Assuming always self-locking
Check γ vs φ; multi-start worms usually are not.
Ignoring heat
Low efficiency means (1−η) of input becomes heat — lubricate/cool.
Wrong q
Choose standard q for shaft stiffness; d₁ = q·m.
No strength rating
Rate per AGMA 6022/6034 / ISO 14521.
Skipping lubrication
Use AGMA worm oil to avoid scoring.

Assumptions

This calculator makes the following assumptions to provide estimates:

Standard geometry: Axial module m common to worm and wheel.
Friction: μ default 0.05 (steel/bronze, lubricated).
90° axes: Worm and wheel axes perpendicular.
Efficiency model: η = tanγ/tan(γ+φ) (worm driving).
No rating: Bending/wear capacity is separate.

Limitations

Be aware of these limitations when using this calculator:

No strength rating: AGMA/ISO capacity not computed.
Heat not sized: Cooling/lube only noted.
Single stage: One worm + wheel only.
Ideal friction: Constant μ assumed.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a worm gear?
A worm gear is a screw (the worm) meshing with a toothed wheel (the worm wheel). It gives large reductions in a single stage and can self-lock so the wheel cannot back-drive the worm.
How is worm gear ratio calculated?
Ratio i = z₂ / z₁, where z₂ is wheel teeth and z₁ is worm starts. A single-start worm with 40 teeth gives 40:1; a double-start gives 20:1.
What is the lead of a worm?
Lead L = π·m·z₁ is the axial advance per full turn of the worm. For a single start L = π·m (pitch); multi-start worms have larger lead.
What is the lead angle?
Lead angle γ = atan(z₁/q) is the helix angle of the worm thread; it controls efficiency and self-locking. Larger γ → higher efficiency, lower chance of locking.
What is the diameter quotient q?
q = d₁/m ties the worm pitch diameter to the module. Standard q values (8, 10, 12.6, 14, 16) avoid too-thin worm shafts; d₁ = q·m.
What is the center distance?
a = (d₁ + d₂)/2 = m·(q + z₂)/2, the distance between the worm and wheel axes.
What is worm gear efficiency?
η = tan γ / tan(γ + φ), where φ = atan μ is the friction angle. Single-start worms run ~30–50% efficient; multi-start with large γ reach 70–90%.
When is a worm gear self-locking?
When the lead angle γ ≤ the friction angle φ (i.e. γ ≤ atan μ). Then the wheel cannot drive the worm, giving a braking (holding) effect.
Why use a worm gear?
Compact high reductions (5:1 to 100:1 in one stage), quiet right-angle drive, self-locking option, and smooth operation.
What are the disadvantages?
Lower efficiency than spur/helical (more heat), higher wear, the wheel often needs bronze, and self-locking units lose most of the input power as heat.
What materials are used?
The worm is hardened steel; the wheel rim is usually bronze (phosphor or aluminium bronze) for its anti-galling and run-in properties.
What is a single vs double start worm?
Starts = threads on the worm. More starts raise the ratio per wheel tooth and the lead angle (more efficient, less likely to lock).
What is the axial module?
m is the module measured in the worm axis plane; the wheel uses the same m so the pair is conjugate.
How do I choose q?
Lower q gives a smaller worm but weaker shaft; higher q stiffer but bigger. Standard values keep the worm from being too slender. q also sets γ via tan γ = z₁/q.
What is the pitch line velocity?
v = π·d₁·n₁/60 (m/s with d₁ in m). It scales heat and sets the AGMA speed zone (lower zones for higher reduction).
What is the output speed?
n₂ = n₁ / i. A 1500 rpm input and 40:1 gives 37.5 rpm at the wheel.
What is the output torque?
T₂ = T₁ · i · η. The ratio multiplies torque but efficiency (often < 1) eats part of the gain.
What is the difference between worm and helical gears?
Worm meshes a screw with a wheel at 90° offset axes; helical meshes parallel-offset gears. Worm allows much higher single-stage ratio and self-locking.
What is back-driving?
Driving the wheel to turn the worm. Possible only when γ > φ (not self-locking). Self-locking units prevent back-driving.
What is the AGMA standard?
AGMA 6022 / 6034 cover worm geometry and rating (bending and wear/pitting).
What heat problems occur?
Low-efficiency worms dissipate most input as heat; large reductions need cooling or forced lubrication to avoid scoring.
What lubrication is needed?
High-viscosity EP worm oil (AGMA 7 or 8 comp) forming an EHL film; splash or forced, sometimes cooled.
What is the thrust in a worm set?
Both worm and wheel see axial and radial loads from the thread helix; bearings must take combined thrust.
Can worm gears be reversible?
Only if not self-locking (γ > φ). Otherwise they are one-way (worm drives wheel).
What is the maximum ratio?
Single-stage up to ~100:1 (z₂≈100, z₁=1); beyond that use two stages.
What is the minimum wheel teeth?
About 28–30 to avoid undercut and keep the wheel concave; very low ratios use fewer.
What is the cost?
Worm+bronze wheel pairs run $100–$2000 depending on size and bronze; bigger or precision units more.
What are typical applications?
Conveyors, lifts, tuning drives, valve actuators, winches, presses, and any place needing a self-locking right-angle reduction.
How do I improve efficiency?
Use more starts (higher γ), lower friction (better lube/material), and avoid over-small q; but efficiency gains reduce self-locking.
What is the contact pattern?
A thin marked band on the wheel flank; center it by adjusting center distance and thrust position.
Does the calculator include efficiency?
Yes — η = tanγ/tan(γ+φ) with μ (default 0.05), and it flags self-locking when γ ≤ φ; output torque uses T₁·i·η.
What friction coefficient should I use?
0.04–0.06 for well-lubricated steel/bronze (default 0.05). Worn or dry contacts are higher, lowering η and increasing locking tendency.
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Glossary

Technical terms used in mortgage calculations explained in plain language.

Lead (L)
Axial advance per worm turn = π·m·z₁.
Lead Angle (γ)
Thread helix angle; tan γ = z₁/q.
Diameter Quotient (q)
d₁/m; standard 8–16.
Center Distance (a)
(d₁+d₂)/2 = m(q+z₂)/2.
Friction Angle (φ)
atan μ; locking when γ ≤ φ.
Starts (z₁)
Worm thread count; 1–4.
Self-locking
Wheel cannot back-drive the worm.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AGMA 6022AGMA 6034ISO 14521DIN 3975

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Page Information

Last Updated:

August 2026

Version:

2.1.0

License:

Free to use for educational and professional purposes.