Sprocket pitch diameter, explained

A roller chain cannot bend mid-link, so it never actually follows a circle — it sits on the sprocket as a polygon of rigid chords, one chain pitch long each. The pitch diameter is simply the circle that polygon's corners (the roller centers) lie on, and that single fact produces the formula, the odd decimals, and the 17-tooth rule.

The formula is a chord relation

N rollers spaced around a circle subtend 360°/N each; the chord between neighbors must equal the chain pitch p:

p = PD · sin(180°/N) → PD = p / sin(180°/N)

For a 17-tooth sprocket on #40 chain (p = 1/2 in): PD = 0.5 / sin(10.59°) = 2.721 in. The sprocket pitch diameter calculator runs this with chain speed and travel per revolution; the full-drive version (ratio, both sprockets) is the sprocket calculator.

Each tooth adds p/π

For more than a handful of teeth, sin(180°/N) ≈ 180°/N, so PD ≈ N·p/π — the circumference of the pitch circle is nearly N pitches laid along an arc. Going from 17 to 18 teeth on #40 chain grows the PD by 0.158 in, almost exactly p/π = 0.159 in. Handy both ways: estimate any sprocket's PD in your head (N × 0.159 in for #40), or count how many teeth fit a diameter budget.

Chordal action: the 17-tooth rule

Because the chain rides chords, the effective radius alternates between the full pitch radius and its cosine projection every tooth — the chain speeds up and slows down N times per revolution:

speed variation = 1 − cos(180°/N)

That is 6.0% at 9 teeth, 4.1% at 11, 1.7% at 17, 0.8% at 25. The pulsation shakes the chain, hammers the joints and makes small sprockets loud at speed — which is why design guides push 17+ teeth for the driver at meaningful RPM, and why the calculator's verdict flags below that. (Small sprockets are fine for slow, manual or space-starved drives; the penalty scales with speed.)

Worked example — a 17:40 drive

#40 chain, 17-tooth driver at 300 RPM into a 40-tooth driven sprocket:

PD₁ = 2.721 in · PD₂ = 6.373 in · ratio = 40/17 = 2.35 · chain speed = 17 × 0.5 × 300 / 12 ≈ 213 fpm

Note the ratio comes from the tooth counts alone — exact, by construction — while the pitch diameters are what set shaft spacing, clearance and chain length. Chain pull and the tension margin for the same drive live in the chain pull calculator and roller chain tension calculator.

Identifying an unknown sprocket

PD is invisible to calipers — the practical route is the outside diameter, which catalogs cut to approximately OD ≈ p·(0.6 + cot(180°/N)). The 17-tooth #40 sprocket mics about 2.97 in over the tips. So: count teeth, measure OD, solve for which standard pitch fits (the same 17 teeth in #60 chain would give PD = 4.08 in and a proportionally larger OD), then confirm against the chain you actually have.

Common mistakes

• Using the OD as the pitch diameter. The tips stand well above the pitch circle (2.97 vs 2.72 in for the 17T example) — center-distance and chain-length math done on OD comes out long.
• Computing ratio from diameters. Tooth counts give the exact ratio; diameters re-derive it with rounding error. Use N₂/N₁.
• Ignoring chordal action on a fast small sprocket. An 11-tooth driver pulses 4% at every tooth engagement — fine on a hand-cranked feed, miserable at 1,800 RPM.
• Mixing chain series. PD scales linearly with pitch: the same tooth count is 2.72 in on #40 and 4.08 in on #60. Verify pitch with a rule across several links before trusting any diameter math.

Frequently asked questions

How do you calculate sprocket pitch diameter?

PD = p / sin(180°/N) — chain pitch divided by the sine of the half-tooth angle. A 17-tooth sprocket for #40 chain (1/2 in pitch): 0.5 / sin(10.59°) = 2.721 in. It is geometry, not a catalog convention: the chain pitches must sit as chords of the pitch circle.

Why is sprocket pitch diameter not a round number?

Because the pitch circle is defined by N chords of fixed length p, and p/sin(180°/N) is irrational for any practical tooth count. Each added tooth grows the PD by almost exactly p/π (0.159 in for 1/2 in chain) — the circle the chain wants, but never a catalog-friendly number.

Why do sprockets have a minimum of about 17 teeth?

Chordal action. The chain rides chords, not the circle, so its speed pulses by (1 − cos(180°/N)) every tooth: 6% at 9 teeth, 4% at 11, 1.7% at 17, 0.8% at 25. Below ~17 teeth that pulsation becomes audible roughness, vibration and accelerated wear at speed.

How do you measure the pitch diameter of an existing sprocket?

You usually measure the outside diameter and work backwards: catalogs cut teeth so OD ≈ p × (0.6 + cot(180°/N)). A 17-tooth #40 sprocket measures about 2.97 in over the tips while its true PD is 2.72 in. Count the teeth, measure OD to identify the chain series, then compute PD.