Sprocket dimensions: pitch diameter, outside and root diameter, tooth form and a pitch diameter chart

A sprocket has three diameters that matter: the pitch diameter, where the roller centres run; the root diameter, at the bottom of the tooth gaps; and the outside diameter, over the tooth tips. The first two decide whether the chain fits; the third is only where the teeth are cut off. This guide gives the formulas, a worked example for an ANSI 40 sprocket, a chart of pitch diameters by tooth count and the tooth form the standards draw.

The sprocket generator does the sums for any chain and tooth count and draws the profile as a DXF; the pitch and roller of each chain are in the roller chain size chart.

The formulas

Pitch diameter

d = p / sin(180° / z) p is the chain pitch and z the number of teeth. The chord between two neighbouring roller centres is exactly one pitch.

Root diameter

df = d − d1 d1 is the roller diameter, or the bushing diameter on the rollerless 25 and 35. On an even number of teeth it is measured straight across two opposite gaps.

Caliper reading, odd number of teeth

dc = d × cos(90° / z) − d1 With an odd number of teeth no gap is opposite another, and this is the reading across the two gaps nearest to opposite.

Outside diameter

ANSI: da = p × (0.6 + cot(180° / z)) ; ISO 606: from d + p (1 − 1.6 / z) − d1 to d + 1.25 p − d1 The outside diameter does not touch the chain: the standards give it as a range, or as the value for a given cutter.

Worked example: ANSI 40 chain, 20 teeth

ANSI catalogues print 3.196, 2.883 and 3.457 inches for this sprocket; the small differences are the truncated third decimal. The generator cuts the tip in the middle of the ISO range, 87.04 mm, because its tooth sits in the middle of the ISO envelope.

Pitch diameter chart

The pitch diameter is the pitch times a factor that depends only on the number of teeth, 1 / sin(180° / z). Multiply the factor by your pitch, or read the half-inch columns for a 40, 41, 08B, 415, 420, 428 or bicycle chain.

The tooth form

ISO 606 draws each gap as a roller seat -- an arc a little larger than the roller, centred on the pitch circle -- and two convex flank arcs tangent to it, cut off at the outside diameter. It gives two limiting gaps, a tighter and a looser one, and accepts any flank in between. The ANSI form of ASME B29.1 builds the gap from a seating curve, a working curve and a topping curve, and is the shape of hobbed catalogue sprockets.

Tighter gap (ISO 606)

ri = 0.505 d1 ; α = 140° − 90° / z ; re = 0.12 d1 (z + 2) ri is the seat radius, α the seat angle and re the flank radius, with d1 in millimetres.

Looser gap (ISO 606)

ri = 0.505 d1 + 0.069 ∛d1 ; α = 120° − 90° / z ; re = 0.008 d1 (z² + 180) The cube-root term only works in millimetres.

For thermal cutting the middle of that envelope is the sensible nominal: plasma and laser wander a few tenths of a millimetre either way, and a nominal on one limit throws half that error outside the standard. It is what the generator draws.

Cutting a sprocket from plate

1. Choose the plate thickness from the chain's inner width: the tooth has to pass between the inner plates with clearance.
2. Cut the bore first, undersize, and the outline last, so the part is still held by the sheet while the profile is cut.
3. Let it cool flat on the table; a hot plate pulls out of flat and the teeth wobble.
4. Finish the bore on the lathe, centred on the root diameter, so that the teeth run true.
5. Check the root diameter, or the caliper reading on an odd count, and fit the chain over a full turn.

Common mistakes

• Checking the sprocket by its outside diameter: it changes with the cutter and the standard; the root diameter is the check.
• Reading the odd-tooth caliper measurement as the root diameter: on 17 teeth for a 40 chain they are 60.90 and 61.19 mm.
• Too few teeth on a driving sprocket: below 17 the chain speeds up and slows down on every tooth.

Frequently asked questions

How do you calculate the pitch diameter of a sprocket?

Divide the chain pitch by the sine of 180° over the number of teeth: d = p / sin(180° / z). A 20-tooth sprocket for a half-inch chain: 12.7 / sin 9° = 81.18 mm.

What is the root diameter of a sprocket?

The diameter at the bottom of the tooth gaps: the pitch diameter minus the roller diameter. On an even number of teeth it is measured across two opposite gaps.

How do I measure a sprocket with an odd number of teeth?

Across the two gaps nearest to opposite, with callipers: d × cos(90° / z) − d1. For a 17-tooth sprocket on a 40 chain that is 60.90 mm.

What is the outside diameter of a 20-tooth #40 sprocket?

3.457 in, 87.80 mm, by the ANSI formula that catalogues print. ISO 606 allows 84.94 to 89.13 mm for the same sprocket.