August 2026 13 min read

Bevel Gear Calculator

Free bevel gear calculator. Compute pitch diameters, pitch (cone) angles, outer cone distance, face-width limits and gear ratio for straight, spiral and hypoid bevel gears on intersecting shafts, plus output speed/torque with efficiency.

Bevel Gear Calculator

Bevel Gear Inputs

Unit System

SI — Module (mm) Imperial — DP

Module m (mm)

Tooth size

Pinion Teeth z₁

Gear Teeth z₂

Shaft Angle Σ (°)

Usually 90°

Bevel Type

Sets efficiency

Face Width (optional) (mm)

Leave blank for default

Input Speed (rpm)

Optional

Input Torque (N·m)

Optional

Auto-calculate instantly

Gear Ratio

2 : 1

δ₁ = 26.6°, δ₂ = 63.4°.

Geometry is valid; check face width and virtual teeth before rating.

Pinion δ₁

26.57°

Gear δ₂

63.43°

Pitch Dia d₁

60mm

Pitch Dia d₂

120mm

Cone Distance

67.08mm

Max Face Width

22.36mm

Efficiency

97%

Virtual z_v1

22.4

Pinion δ₁

26.57°

Pitch angle

Gear δ₂

63.43°

Pitch angle

Pitch Dia d₁

60mm

m·z₁

Pitch Dia d₂

120mm

m·z₂

Cone Distance

67.08mm

Outer R_e

Max Face Width

22.36mm

min(R_e/3, 10m)

Efficiency

97%

straight

Virtual z_v1

22.4

z₁/cos δ₁

Result Summary

Module	3 mm
Ratio	2 : 1
Shaft Angle	90°
Pinion δ₁	26.57°
Gear δ₂	63.43°
d₁ / d₂	60 / 120 mm
Cone Distance	67.08 mm
Face Width	15.65 mm
Efficiency	97 %
Output Speed	750 rpm
Output Torque	97 N·m

Estimated Result — Uses standard intersecting-shaft bevel geometry: d = m·z, δ₁ = atan(sinΣ/(i+cosΣ)), R_e = d/(2·sinδ), face-width limit min(R_e/3, 10m), and virtual teeth z_v = z/cosδ for strength. Efficiency (straight 97%, spiral 96%, hypoid 92%) is applied to output torque. It does not rate bending/contact strength (use AGMA 2003 / ISO 10300) nor set mounting distance/tolerances.

Interactive Diagram

Pitch angles satisfy δ₁ + δ₂ = Σ; the gear blank is a truncated cone.

Visualizations

Bevel Gear Dimensions (mm)

60mmd₁

120mmd₂

67mmR_e

22.4mmb_max

15.7mmb

Face Width Utilization

b / b_max70%

Formula

Bevel gears run on intersecting shafts. Pitch diameters follow d = m·z at the heel; the pitch (cone) angles come from the shaft angle and ratio; the cone distance and face-width limit size the blank. Virtual teeth z_v = z/cosδ let you use spur-gear strength methods.

d = m · z

Alternative Form

δ₁ = atan(sinΣ / (i + cosΣ)) , R_e = d/(2·sinδ) , b_max = min(R_e/3, 10m)

Variables

Symbol	Name	Unit	Description
m	Module	mm	Pitch diameter per tooth
z₁, z₂	Teeth	—	Pinion and gear teeth
Σ	Shaft angle	°	Angle between shafts
δ₁, δ₂	Pitch angles	°	Cone half-angles; δ₁+δ₂=Σ
R_e	Outer cone distance	mm	Apex to heel
b	Face width	mm	≤ R_e/3 and ≤ 10m
z_v	Virtual teeth	—	z/cos δ

Explanation

Unlike spur gears, bevel teeth lie on a cone, so the pitch diameters are still m·z but the blank shape is set by the pitch angles. For 90° shafts the pinion angle is simply arctan(1/ratio). The outer cone distance fixes how big the blank is, and the face width must stay within a fraction of it so load is shared and the tooth does not taper too much. With a type chosen, the ratio plus efficiency gives output speed and torque.

Derivation

From the cone triangle, tan δ₁ = sin Σ / (z₂/z₁ + cos Σ), i.e. δ₁ = atan(sinΣ/(i+cosΣ)) with i = z₂/z₁; δ₂ = Σ − δ₁. R_e = (m·z)/(2 sinδ). Recommended practice limits b to R_e/3 or 10m. Output speed n_out = n_in/i; output torque T_out = T_in·i·η with η from the gear type.

Step-by-Step Calculation

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

Step 1

Compute pitch diameters from module and teeth.

d = m·z

m:3 mm

z₁:20

z₂:40

Intermediate: d₁ = 3×20, d₂ = 3×40

Result: d₁ = 60 mm, d₂ = 120 mm

Step 2

Find the pitch (cone) angles.

δ₁ = atan(sinΣ / (i + cosΣ)) , δ₂ = Σ − δ₁

Σ:90°

i:2

Intermediate: δ₁ = atan(sin90°/(2+cos90°))

Result: δ₁ = 26.57°, δ₂ = 63.43°

Step 3

Outer cone distance.

R_e = d / (2·sin δ)

d₂:120 mm

δ₂:63.43°

Intermediate: R_e = 120 / (2·sin 63.43°)

Result: 67.08 mm

Step 4

Limit face width.

b_max = min(R_e/3, 10·m)

R_e:67.08 mm

m:3 mm

Intermediate: b_max = min(22.36, 30)

Result: 22.36 mm

Step 5

Output speed and torque (with efficiency).

n_out = n_in/i , T_out = T_in·i·η

η:97 %

Intermediate: η = 97% (straight)

Result: 750 rpm, 97 N·m

Final Answer

2 : 1

Worked Examples

See how the calculator works with different scenarios.

Scenario

3 mm module, 20/40 teeth, 90° straight bevel.

Inputs

m:3 mm

z₁/z₂:20/40

Σ:90°

type:straight

Calculation

δ₁ = atan(sin90/(2+cos90)) = atan(1/2) = 26.57°; δ₂ = 63.43°; d₁ = 60, d₂ = 120 mm; R_e = 120/(2·sin63.43)=67.1 mm; b_max = min(22.4,30)=22.4 mm; η=97%

Result

2:1, δ₁=26.6°, δ₂=63.4°

Interpretation

A standard 90° right-angle reduction; cone geometry sets the blanks.

Understanding Your Result

Your Result: 2 : 1

Typical range: Keep face width at or below 22.36 mm (≤ R_e/3, 10m). Your b = 15.65 mm is within limit.

What This Means

This straight bevel pair has ratio 2:1, pitch angles δ₁ = 26.6° and δ₂ = 63.4°, outer cone distance 67.08 mm, max face width 22.36 mm, at 97% efficiency.

Why It Matters

Bevel geometry dictates whether the two cones mesh at the correct apex; wrong pitch angles or face width cause edge loading, noise and early failure. The ratio and efficiency then set the speed/torque delivered to the crossed shaft.

Professional Interpretation

Use z_v1 = 22.4, z_v2 = 89.4 in the Lewis/Hertz formulas (AGMA 2003 / ISO 10300) for bending and pitting. Hold mounting distance tightly and verify the contact pattern. Output torque ≈ 97 N·m.

Cone angles

δ₁ = 26.57°, δ₂ = 63.43° (Σ = 90°).

Virtual teeth

z_v1 = 22.4, z_v2 = 89.4 — use for strength.

Efficiency

97% (straight).

Recommendations

•Face width is within recommended limits.
•For 90° shafts, miter gears need equal teeth (45° pitch).
•Use spiral (or hypoid) for smoother, higher-capacity drives.
•Rate bending/pitting with virtual teeth per AGMA/ISO.
•Control mounting distance and check the contact pattern.

Educational Content

Introduction

Bevel gears are how machines turn power around a corner. Their teeth sit on a cone, so the whole geometry is about two intersecting pitch cones meeting at a common apex. This calculator takes the module, tooth counts and shaft angle and returns the pitch diameters, pitch (cone) angles, cone distance, face-width limit and ratio — plus output speed and torque when you add a type and input.

Concept Explanation

Pitch diameter is still d = m·z, measured at the heel (large end). The pitch angles follow from the shaft angle Σ and ratio: δ₁ = atan(sinΣ/(i+cosΣ)), δ₂ = Σ−δ₁, so for 90° and equal teeth (miter) both are 45°. The outer cone distance R_e = d/(2·sinδ) sizes the blank, and face width is limited to min(R_e/3, 10m) so the tapering tooth shares load. Virtual teeth z_v = z/cosδ let standard spur strength formulas apply.

Applications

▸Automotive differentials (hypoid)

▸Right-angle industrial gearboxes (spiral)

▸Hand drills and power tools (straight)

▸Helicopter and marine transmissions

▸Machine tools and printing presses

▸Teaching intersecting-shaft kinematics

Benefits

•Returns full cone geometry from module + teeth
•Handles any shaft angle, not just 90°
•Computes ratio, cone distance and face-width limit
•Adds efficiency for output speed/torque
•SI / imperial (module/DP) consistent

Limitations

•No bending/pitting strength rating
•Does not set mounting distance/tolerances
•Assumes standard straight/spirals (no Zerol detail)
•Face width is recommended, not optimized
•Single pair only

Best Practices

•Keep face width ≤ R_e/3 and ≤ 10m
•Use virtual teeth for AGMA/ISO strength
•Hold mounting distance to tight tolerance
•Check the contact pattern with marking compound
•Pick spiral/hypoid for smoothness and capacity

Common Misconceptions

Myth: Bevel pitch angle is always 45°.

Truth: Only for 1:1 miter at 90°; otherwise δ₁ = atan(sinΣ/(i+cosΣ)).

Myth: Face width can be anything.

Truth: It must stay within R_e/3 and 10m or load is uneven.

Myth: Use spur formulas directly.

Truth: Use virtual teeth z_v = z/cosδ first.

Myth: All bevels are 90°.

Truth: Shaft angle can be any value; the formula generalizes.

Practical Tips

•Ratio 2:1, cone distance 67.08 mm.
•Face width is within recommended limits.
•Use virtual teeth z_v = z/cosδ for strength checks.
•Spiral/hypoid run smoother but need thrust bearings.
•Rate bending/pitting per AGMA/ISO after geometry.

Common Mistakes

Wrong pitch angles

Use δ₁ = atan(sinΣ/(i+cosΣ)), not a fixed 45°.

Face width too large

Cap at min(R_e/3, 10m) for even load.

Using z instead of z_v

Strength uses virtual teeth z/cosδ.

Ignoring mounting distance

Mesh position depends on tight mounting distance.

Skipping strength rating

Rate per AGMA 2003 / ISO 10300.

Assumptions

This calculator makes the following assumptions to provide estimates:

Standard tooth: Full-depth bevel proportions assumed.

Valid shaft angle: 0 < Σ < 180°.

Efficiency by type: 97/96/92% for straight/spiral/hypoid.

Face width optional: Defaults to 0.7·b_max if blank.

No strength rating: AGMA/ISO capacity is separate.

Limitations

Be aware of these limitations when using this calculator:

No strength: Bending/pitting needs AGMA/ISO 10300.

No mounting distance: Tolerance/layout out of scope.

Single pair: One intersecting-shaft pair.

Face width guidance: Recommended, not optimized.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a bevel gear?

A bevel gear has conical tooth surfaces on a conical blank, transmitting power between intersecting shafts — usually 90°, but any shaft angle is possible. Found in differentials, right-angle drives and hand tools.

How are bevel pitch diameters calculated?

As for spur gears: d = m·z, measured at the large (heel) end of the tooth. The pitch diameter sets the gear size and the center distance.

What are pitch angles in bevel gears?

Pitch angles δ define the cone angles. For a shaft angle Σ: δ₁ = atan(sin Σ / (ratio + cos Σ)), δ₂ = Σ − δ₁. At Σ = 90° and z₁ = z₂ (miter), both are 45°.

What is outer cone distance?

Outer cone distance R_e is the distance from the cone apex to the heel end along the cone: R_e = d / (2·sin δ). It sets the blank size and the face-width limit.

What is face width in bevel gears?

Face width b is the tooth width along the cone. Keep it ≤ R_e/3 and ≤ 10·m (whichever is smaller); too wide causes uneven load from the tapering tooth.

What is mounting distance?

The axial distance from the gear back face to the shaft-centerline intersection. It must be held tightly so the teeth mesh at the correct cone position.

What types of bevel gears exist?

Straight (simple, noisier), spiral (curved teeth, smoother, higher capacity), Zerol (zero spiral), hypoid (offset pinion, used in diffs) and miter (1:1, 45° pitch).

What is the difference between straight and spiral bevel gears?

Straight teeth engage all at once; spiral teeth engage gradually, giving quieter, higher-capacity operation at speed, at the cost of axial thrust.

What is the minimum number of teeth?

About 16–17 at 20° pressure angle for straight bevels; spiral bevels can go as low as 12 thanks to the spiral overlap.

How is bevel gear ratio calculated?

i = z₂/z₁ = d₂/d₁ = sin δ₂/sin δ₁. For 90° shafts this is cot δ₁. It sets the speed/torque relationship.

What are miter gears?

Bevel gears with a 1:1 ratio (equal teeth) and 45° pitch angles for 90° shafts — they redirect rotation without changing speed or torque.

What is a hypoid gear?

A spiral-bevel variant with the pinion axis offset from the gear axis, allowing a larger pinion and smoother mesh; standard in automotive differentials but with lower efficiency.

What is the efficiency of bevel gears?

Straight ≈ 96–98%, spiral ≈ 95–97%, hypoid ≈ 90–95% (extra sliding from the offset eats efficiency, especially at high ratios).

What materials are used for bevel gears?

Case-carburized steel (8620, 4320) for high load, through-hardened (4140) medium, nitrided for wear. Auto diff gears are typically carburized.

How are bevel gears manufactured?

Straight bevels by planers/generators (Gleason, Klingelnberg); spiral bevels need complex dedicated machines; precision units are ground.

What is the Tredgold approximation?

It replaces a bevel gear with an equivalent (virtual) spur gear at the mean cone distance so standard spur strength formulas can be used.

What is the virtual number of teeth?

z_v = z / cos δ. Used to pick the Lewis form factor and to check undercut, because the tooth behaves like a spur of z_v teeth.

How do I calculate bevel torque capacity?

Use the modified Lewis (bending) and Hertz (contact) formulas with the virtual spur parameters at the mean cone distance.

What is the shaft angle?

The angle Σ between the intersecting shafts; usually 90° but any value is possible, with δ₁ + δ₂ = Σ always.

What is backlash in bevel gears?

Clearance between mating teeth at the pitch cone, typically 0.05–0.20 mm for precision sets, set by adjusting mounting distance.

How do I set up bevel gears?

Set both gears at correct mounting distance, adjust backlash axially, check the contact pattern with marking compound, and preload the bearings.

What is the tooth contact pattern?

The marked area of flank contact under light load; a good pattern is centered and covers 50–70% of the flank. Edge contact means misalignment.

What causes bevel gear noise?

Wrong mounting distance, excess backlash, poor contact pattern, runout and vibration. Spiral and ground gears are quieter.

How do I lubricate bevel gears?

High-viscosity EP oils (GL-5 for hypoids), splash or forced circulation. The sliding/rolling contact needs a strong EHL film.

What heat treatment is used?

Case carburizing (HRC 58–62 surface) is common; through-hardening for large gears; nitriding where distortion must be minimal.

What are bevel gear failures?

Bending fatigue (root cracks), pitting (flank), scuffing (lube failure), overload fracture and wear. Good material/lube prevents most.

What is the AGMA standard for bevel gears?

AGMA 2005-D03 (geometry), AGMA 2003-B97 (rating), AGMA 2009-B01 (tolerances).

What is the cost of bevel gears?

More than spur gears: straight pairs $50–500, spiral $200–2000, hypoid sets $100–1000 (auto), precision ground $1000–10000.

Can bevel gears speed up as well as reduce?

Yes — they work in either direction, though reducers are more common. Step-up efficiency is slightly lower.

What is the maximum ratio for straight bevels?

About 5:1; beyond that the pinion undercuts and the gear grows large. Spiral bevels reach ~8:1; higher needs multiple stages.

How does spiral angle affect performance?

30–35° spiral gives gradual engagement (quiet, high capacity) but more axial thrust; direction sets thrust direction.

What is the difference between bevel and miter gears?

Miter is a 1:1 bevel (45° at 90°). All miters are bevels, but bevels can have any ratio.

How do I design a bevel pair?

Fix power/speed/ratio/shaft angle, choose module/teeth/pressure angle, find pitch diameters and angles, set cone distance and face width, rate by the virtual-spur method, then specify mounting distance.

What are bevel gear applications?

Auto differentials (hypoid), hand drills (straight), right-angle boxes (spiral), helicopter transmissions, marine, machine tools, printing presses.

Does the calculator include efficiency?

Yes — straight ≈ 97%, spiral ≈ 96%, hypoid ≈ 92% are applied to the output torque, with the ratio setting output speed.

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Related Guides

Deepen your understanding with these educational resources.

AGMA — Bevel gear standards

Bevel gear geometry and rating.

Gleason — Bevel gear technology

Bevel/hypoid cutting and design.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Pitch Angle (δ): Cone half-angle of the gear blank.
Outer Cone Distance (R_e): Apex-to-heel distance along the cone.
Face Width (b): Tooth length along the cone; ≤ R_e/3 and ≤ 10m.
Shaft Angle (Σ): Angle between the intersecting shafts (usually 90°).
Virtual Teeth (z_v): z/cos δ; equivalent spur tooth count.
Miter Gear: 1:1 bevel gear, 45° pitch at 90°.
Mounting Distance: Back-face to shaft-axis intersection.
Hypoid: Bevel with offset pinion axis.

Authoritative References

Trusted sources and standards referenced by this calculator.

AGMA — Bevel gear standards ISO 10300 — Bevel gear rating Gleason — Bevel gear technology MIT OpenCourseWare — Gear design KHK Gears — Bevel gear catalog

Applicable Standards

AGMA 2005-D03AGMA 2003-B97ISO 10300Gleason / Klingelnberg

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Page Information

Last Updated:

August 2026

Version:

2.1.0

References:

License:

Free to use for educational and professional purposes.