Change Gear Calculator

Most machine tools have at least one gearbox, but the ability to set the gear ratio may vary wildly. Simple ratios, i.e. changing between "High/Low" or "Inch/Metric" might be possible by flipping a lever, or engaging one gear. But in gear-making equipment, like hobbers, shapers, and grinders, an exact ratio is required to make a gear with the right number of teeth. These ratios use four gears arranged (A/B)*(C/D) to give the machine the correct gear ratio. These change gear calculations can be found in the Machinery's Handbook or tables from the OEM, or generated with this change gear calculator. The machine itself (hobber, shaper, grinder) will also have an internal Machine Constant, so for a given tooth count, one needs to solve: Machine Constant / Number of Teeth = (A/B)*(C/D)

Our change gear calculator for hobbers, shapers, and grinders, takes the math out of equation and allows you to get a range of solutions from just the Number of Teeth and Machine Constant. The range of solutions grows even more when two of the change gears form a common ratio (i.e. 1:1, 1:2, 2:3). Rather than repeat out every option: 20/20, 21/21, .... the calculator will treat this as 1:1. Likewise a 2;1 ratio is just 2:1, and any pair of gears (20:40, 30:60) will work.

How to Find your Machine Constant

The machine constant for a hobbing machine can be found by installing all four change gears with the same number of teeth, and recording how many times the hob rotates relative to the gear blank. With the machine off, you can usually rotate it by hand to do this for smaller machines. For larger machines that must be energized to make this happen, use any modern camera and some indicators to mark the rotation of the hob and the table in the same frame. Count it by watching your video. That is, with A, B, C, and D all the same, as the workpiece rotates one time, how many times does the hob rotate? This relative rotation count will determine the Machine Constant for your machine.

Range of Acceptable Gears

Not all gears will fit inside your machine. Large gears may not physically fit in the cabinet, or require too high of a center distance. High gear ratios between individuals gears (e.g. 125:21) also make a proper mesh more challenging. Our calculator preferences common ratios (1:1, 1:2, 2:3) and gears that are close in size to give you the most options that will fit your machine. We did not constrain any output data, but we prioritize answers with the smallest physical gear size.

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Making Prime Numbered Gears

Making a gear with a prime number of teeth can be an extra challenge. A prime number, by definition, cannot be broken down to a product of any two numbers other than 1 and itself. That is, the only way to multiply to get 73 is 73*1. Practically, this often means that you'll need the exact change gear to make the gear you want to.

And yet, prime numbered gears remain extremely popular in gearbox design. The same attribute that makes prime numbered gears difficult to machine makes them attractive for evenly distributing wear across a pair of meshing gears. Consider two gears with the same number of teeth, i.e. 40 and 40. Each tooth is always in contact with its specific mate. If one gear had 41 teeth, that gear would need to rotate 40 times before finding its original "tooth mate," distributing any non-uniformities (wear, damage, manufacturing defects, etc.) across the entire set of teeth, not just concentrated on one point.

The prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The Math of Change Gear Calculation

From a math standpoint, the selection of change gears is a problem of continued fractions, and conjugate fraction methods. In reality there are many more boundary conditions, such as: the machine size, limits on the "Scissor", and which change gears the operator actually possesses.

A tabular solution to a very specific set of ratios is in the Pfauter references below – the math was done once and published in booklets. They solve for gear ratios between 0.10000 to 1.0000 and tabularize the resulting change gears (a/b)*(c/d) using change gear sets from 18 to 80 teeth. These books are difficult to find.


  • 1950, Pfauter Wechselräder Tabellen, Becher und Körner (German)

    1964, Pfauter Change Gear Tables, Becher and Koerner (English)

    1971, ASME, A Computer Algorithm to Design Compound Gear Trains for Arbitrary Ratio, Dil Pare

    1981, Machinery's Handbook 21st Edition (1981),  Pages 1435-1465
    (Includes a nice set of logarithm tables for Tooth Counts Between 15-120. Sadly this set of handy tables was removed in the 23rd edition)

    Machinery's Handbook 29th Edition (2012), Page 11 - Continued Fractions
    Page 12 Conjugate Fractions
    (There is a great example calculation on how to manually iterate to a solution)

Here's a Quick Summary:

› Input your machine constant

› (optionally) Include a list of change gears you have

› Change gear combinations are sorted to preference common ratios