Can a Gear Work With Only One Tooth? Single-Tooth Gear Kinematics Explained
- Breno Cruz
- 2 minutes ago
- 3 min read
A gear with just one tooth sounds like it should barely function — one bump, one slot, done. But the moment you actually run the numbers, the result is genuinely counterintuitive, and it exposes something most people get wrong about how gears actually work.
In this mechanism, the red driving gear has the pitch diameter to hold 40 teeth, but 39 of them have been machined away, leaving exactly one. It meshes with a blue driven gear that has 20 full teeth.
Most people's first guess is that one full rotation of the red gear advances the blue gear by 1/20 of a revolution — after all, one tooth engaging one gear should move it by one tooth space, right?
It's actually 3/20.
Here's the kinematics that explains why — plus a free downloadable CAD file of this exact mechanism if you want to model, animate, or 3D print it yourself.
What Is a Single-Tooth Gear Mechanism?
A single-tooth gear is an extreme case of what's generally called a mutilated gear or incomplete gear — a gear where teeth have been deliberately removed from most of the circumference, leaving only a working section behind.
Most mutilated gear designs leave an arc of several teeth intact for indexing and counting mechanisms. This mechanism pushes that idea to its limit: only a single tooth remains, mounted on a gear body sized as if it carried a full 40-tooth set.
It meshes with a completely ordinary 20-tooth spur gear. For the rest of the driving gear's rotation — everywhere except the moment that one tooth passes by — the two gears simply aren't in contact at all; the driven gear sits idle until the tooth comes back around.
Why the Math Isn't What You'd Expect
The instinctive assumption is simple tooth counting: one tooth on the driver should nudge the driven gear by exactly one tooth space, or 1/20 of a revolution. That assumption treats gear meshing like two ratchets clicking past each other — contact happens at a single instant, then releases.
That's not how involute gear teeth actually work.
Real gear teeth don't meet at a single point and separate; they stay in continuous contact across a swept arc of engagement, made up of two parts:
The arc of approach: The portion of rotation before the pitch point, where the driving tooth first makes contact with a driven tooth and pushes it forward as it approaches full mesh.
The arc of recess: The portion of rotation after the pitch point, where the same driving tooth continues pushing the driven tooth forward as it separates from mesh.
On a normal gear with a full set of teeth, this is invisible — by the time one tooth finishes its arc of engagement, the next tooth has already picked up the load. But on a single-tooth gear, there's no next tooth to hide the effect.
The one tooth present has to do its entire job — both the approach and the recess — in isolation, and because it's sized for a 40-tooth pitch diameter rather than a fine-pitched gear, its profile sweeps across a wide enough arc to physically push the driven gear through three tooth spaces, not one, before finally losing contact.
The Motion, Step by Step
Idle phase: For most of the driving gear's rotation, the toothless arc rotates past the driven gear with no contact — the driven gear stays stationary.
Engagement begins (arc of approach): The single tooth swings into range, contacts the first available tooth space, and begins pushing forward.
Peak mesh: The tooth passes through the pitch point at its deepest, most direct contact.
Continued push (arc of recess): Past the pitch point, the tooth stays in contact and keeps driving as it withdraws.
Disengagement: By the time the tooth clears the driven gear entirely, the driven gear has advanced three full tooth spaces — 3/20 of a revolution.
Idle phase resumes: The driven gear holds its new position until the tooth comes back around.
Download the CAD File
Grab the CAD files (it’s free)
Frequently Asked Questions
Why doesn't one tooth just advance the driven gear by one tooth space?
Because gear teeth stay in continuous contact through an arc of approach and an arc of recess — a tooth sized for a 40-tooth pitch diameter sweeps a wide enough arc to push the driven gear through multiple tooth spaces before it disengages.
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