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CALCULATORS / GEARS AND GEAR TRAINS

Bevel-gear geometry calculator

Right-angle ideal bevel pair; pitch cone angles only.

gears-bevel.napkin / editable
✓ checks pass
RIGHT-ANGLE BEVEL PAIRPinion 20 teeth · δ 26.565°Gear 40 teeth · δ 63.435°Cone distance 44.721 mmAxes at 90° · pitch-cone schematic

Dashed circles mark pitch geometry. Tooth outlines are schematic, not manufacturing profiles. Labels come from the evaluated .napkin model.

GEARS / TOOTH SCHEMATICSAME SOURCE

Change an input to see the consequences.

cone distance budget44.72 mm✓ passes
whole pinion0.00 ✓ passes
positive pinion20.00 ✓ passes
whole gear0.00 ✓ passes
positive gear40.00 ✓ passes

BASELINE INPUTS

CALCULATED

pinionCone26.565 deg
gearCone63.435 deg
coneDistance44.721 mm

Edits change the baseline source.

Calculated from your source · not saved
Baseline · Baseline
cone distance budget44.72 mm ✓ passes
whole pinion0.00 ✓ passes
positive pinion20.00 ✓ passes
whole gear0.00 ✓ passes
positive gear40.00 ✓ passes
alternative · alternative
cone distance budget43.08 mm ✓ passes
whole pinion0.00 ✓ passes
positive pinion16.00 ✓ passes
whole gear0.00 ✓ passes
positive gear40.00 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

pinion cone
pinionCone = atan(pinion/gear) [deg]
gear cone
gearCone = 90 deg-pinionCone [deg]
cone distance
coneDistance = module*sqrt(pinion^2+gear^2)/2 [mm]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

pinion
20
gear
40
module
2 mm

Calculated

pinion cone
26.5651 deg
gear cone
63.4349 deg
cone distance
44.7214 mm

Assumptions and limits

Right-angle ideal bevel pair; pitch cone angles only.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: Bevel-gear geometry. Inputs: shaft angle; ratio; module. Outputs: pitch-cone angles; nominal envelope. Model scope: Rigid involute-gear kinematics; qualified stress approximations. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

  • Inputs: shaft angle, ratio, module
  • Outputs: pitch-cone angles, nominal envelope
  • Modes: calculate, compare, sweep
  • Drawing: Animated gear assembly

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS

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