Bevel-gear geometry calculator

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

gears-bevel.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	pinionConedeg	gearConedeg	coneDistancemm
	26.5651	63.4349	44.7214
	21.8014	68.1986	43.0813

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

cone distance budget44.72 mm✓ passes

BASELINE INPUTS

pinion20 gear40 module2 mm

CALCULATED

pinionCone26.565 deg

gearCone63.435 deg

coneDistance44.721 mm

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Bevel-gear geometry
# Scope: Right-angle ideal bevel pair; pitch cone angles only.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

pinion = 20
gear = 40
module = 2 mm

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

check cone distance budget: coneDistance <= 50 mm

scenario alternative: pinion = 16

Calculated from your source · not saved

Baseline · Baseline

cone distance budget44.72 mm ✓ passes

alternative · alternative

cone distance budget43.08 mm ✓ 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