Sprocket pitch diameter

What does this quantity describe?

A sprocket's pitch diameter PD is the diameter of the circle through the centers of all engaged chain pins. It follows from PD = p / sin(180°/N), where p is chain pitch and N is tooth count. The formula follows directly from the geometry of a regular N-sided polygon whose vertices are the pin centers and whose side length equals the pitch p.

A sprocket behaves geometrically like a regular polygon with N corners and side length p — not like a smooth circle. The more teeth a sprocket has at a given pitch, the closer this polygon approaches a circle and the smoother the chain runs; with few teeth, the so-called polygon or chordal action becomes noticeable, where chain speed varies slightly within each pitch. Martin Sprocket & Gear's Sprocket Engineering Data catalog uses this exact formula to tabulate pitch diameters for all standard sprockets.

Formula and variables

PD = p / sin(180°/N)

• PD = p / sin(180°/N)
• N = 180° / arcsin(p / PD)

Choose the inputs correctly

Enter two of the three quantities to find the third. Use the standard chain pitch for p (e.g. 19.05 mm for ANSI 60), not center distance or a measured approximation. Tooth count N must be a whole number; a calculated N should always be rounded to the nearest integer.

How to use the calculator

Select the target quantity and enter the two known values with units. Always round a calculated tooth count to a whole number, then check the pitch diameter that actually results from it.

Worked example

Given an ANSI 60 chain with p = 19.05 mm and a sprocket with N = 17 teeth, substitution gives PD = 19.05 / sin(180°/17) ≈ 103.67 mm.

Understand the result and units

A pitch diameter of about 103.7 mm describes the circle along which the engaged chain pins travel — it sits noticeably below the sprocket's actual outside diameter, which additionally accounts for tooth height.

Chain pitch p and pitch diameter PD are given in the same length unit, usually millimetres. N is a dimensionless, integer tooth count.

Typical applications

The formula is used when designing new sprockets, checking whether a sprocket with a given tooth count fits an existing envelope, and converting between tooth count and actual shaft spacing for very short chain drives.

Assumptions, limits and common mistakes

The formula describes the ideal ANSI B29.1 tooth form without wear or manufacturing tolerances. A sprocket's actual outside diameter always exceeds the pitch diameter calculated here because of tooth height; a separate formula or manufacturer table applies for the outside dimension. In practice, at least about 15 to 17 teeth are recommended for a driving sprocket to avoid excessive polygon effect and chain wear.

Common mistake: Do not confuse pitch diameter with a sprocket's outside diameter — for clearance checks, the larger outside diameter is what matters. Also, always round a calculated tooth count to a whole number, since sprockets cannot have fractional tooth counts.