Bevel gears transmit rotation between intersecting shaft axes. Enter the tooth counts, shaft angle and input speed to explore the pitch-cone arrangement and output speed.
Bevel Gears Interactive Calculator
Change the tooth counts and shaft angle to reshape the gear pair; change input speed to update output speed and animation.
Equation Used
- Both shafts intersect at a common pitch apex.
- Tooth-count ratio gives ideal speed ratio without slip.
- Shaft-speed signs depend on the chosen local axis directions; the displayed output is a magnitude.
- Pitch-cone calculations do not determine manufactured tooth compatibility or strength.
Tooth counts and shaft angle determine the pitch cones; input speed determines their rotation. At zero input speed, both gears stop. Crown and obtuse cone combinations are shown with a warning because they fall outside the ordinary external-bevel pair illustrated.
Two pitch cones with one apex
A bevel pair transfers rotation across intersecting shaft axes. Its pitch cones meet along a generator and share an apex at the shaft intersection. Teeth occupy a band of those cones. The blue pinion and gold driven gear in the animation use the entered tooth counts and the calculated cone angles.
The illustration shows simplified straight tooth ribs, a common apex and separate shaft markers. It is a kinematic drawing rather than a tooth-cutting model. KHK’s catalog distinguishes straight and spiral bevel products and provides matched-pair and assembly guidance.
Changing the direction of a drive
Intersecting-axis bevel drives occur in machinery where shafts meet at an angle. An equal-tooth right-angle pair is a miter pair. Hypoid gears have offset axes and require another geometry model; the equations on this page do not size them.
Ratio and pitch-cone calculation
With pinion tooth count Np and driven tooth count Ng, i = Ng / Np and output-speed magnitude = input-speed magnitude / i.
For shaft angle Σ, the pinion cone angle is γp = atan2(sin Σ, i + cos Σ), followed by γg = Σ − γp. Thus sin γg / sin γp = i and the pitch radii have the tooth-count ratio. At a right angle, tan γp = 1 / i.
The use of atan2 matters for large shaft angles. Some permitted combinations reach a 90° pitch angle or pass beyond it. The numerical cone construction remains visible, but the page flags it as outside its ordinary external-bevel illustration. A numerical result is not proof of tooth clearance, manufacturability or acceptable loading.
Worked right-angle example
For 12 pinion teeth, 36 driven teeth and a 90° shaft angle, the ratio is 3:1. The pinion pitch angle is 18.43° and the driven pitch angle is 71.57°. At 200 rpm input, the output magnitude is 66.7 rpm. The same equation gives zero output at zero input.
These values depend only on the entered ratio and axis geometry. Module, face width, tooth system and material are still needed to choose a real matched gear pair.
Scope of this pitch-geometry tool
The model does not estimate efficiency, tooth stress, bearing reactions, noise, contact pattern or service life. KHK’s assembly guidance addresses mounting distance, shaft alignment and gear-specific contact checks; this page does not replace those checks with a universal speed or backlash threshold.
Bevel-gear questions
Why does changing tooth count change the drawing?
The ratio determines pitch-cone angles for the specified shaft angle. It also changes the displayed tooth count and output-speed ratio.
Can output speed exceed input speed?
Yes. A driven gear with fewer teeth than the input gear produces a speed increase in this ideal model.
What happens beyond a 90° pitch-cone angle?
The numerical formula describes a crown or obtuse cone construction. Those combinations lie outside the ordinary external pair illustrated here and need a different tooth geometry.
Does this select an interchangeable manufactured pair?
No. Tooth system, module, mounting dimensions and compatible mating geometry must also match.
Manufacturer reference
KHK Bevel Gears technical information provides photographs of conical straight and spiral gears, mating-pair guidance and assembly/contact cautions. The animation uses simplified geometry to explain the pitch-cone calculation.
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