Why is there pi in my gearing?

In metric gearing there are two methods employed to handle the design for circular pitch gearing: The first works exclusively for rack and pinions. The second is to choose a module that accounts for pi in the calculations.


When I was growing up, there was a phrase that said, “It’s as American as baseball, hot dogs, apple pie, and Chevrolet.” The “it” was anything you that you wanted to identify as being an American standard or custom.

One such American standard is diametral pitch gearing. AGMA, the American Gear Manufacturer’s Association, is the authoritative voice for diametral pitch gearing and they maintain the standards to which all inch pitch gearing is measured. For the past two decades the American gear standards have evolved in a manner that more closely aligns them to the international gear standards set by ISO, JIS, and DIN. Regardless of the standard used, the key dimension in all gearing is the pitch diameter.

The pitch diameter, also known as the reference diameter, is the diameter of the pitch circle. The pitch circle, also known as the reference circle, is the location at which the teeth of two meshed gears will operate. This intersection of these two circles is the point of contact (Figure 1).

Figure 1.

In the American system, gears are measured using diametral pitch. Diametral pitch, commonly abbreviated as DP, is calculated using a unit circle. The value of DP identifies how many teeth exist on a one-inch pitch circle as per the follow formula:

DP = (number of teeth) / pitch diameter

Using common geometry rules, we know that the circumference of a circle is equal to the diameter of the circle multiplied by pi. By substituting the gearing variables, we end up with the following equation:

Pitch Circle = Pitch Diameter * Pi

By dividing the pitch circle value by the number of teeth, you will determine the circular pitch. The circular pitch is also known as the reference pitch.

If we assume that we have a 6DP spur gear with 24 teeth and we run these equations, we get the following results:

As noted in the above calculations, the value of one pitch is 0.5236 inches. For a simple rotary application, this value has zero impact on our application. However, if your application involves the gear translating one inch and then reversing direction and traveling two inches, then calculating the number of teeth to move these distances can be almost impossible. One way to remove this issue from the design is to use a circular pitch gear instead of a diametral pitch gear.

In metric gearing there are two methods employed to handle the design for circular pitch gearing: The first method works exclusively for rack and pinions. It involves using a helical rack and a helical pinion. Since a helical gear has both an axial pitch and a transverse pitch, it is possible to use a standard axial pitch gear with a helix angle of 19 degrees, 31 minutes, and 41 seconds to achieve a transverse pitch that is a whole number.

For example, if you choose a module 3 with a helix angle of 19° 31’ 41”, the resulting circular pitch is 10mm as per the formula:

CP = (Module * PI) / cosine of the helix angle

The second method is to choose a module that accounts for pi in the calculations. This is the method used for straight tooth rack and pinions and other circular pitch applications. An example is as follows:

Using this method, each reference pitch is a fixed value. By rotating the gear in the above example one complete rotation, the resulting linear translation is 250mm. As such, gears produced as circular pitch can be highly effective for pick and place or other repetitive linear motion or positioning applications.

Using traditional inch or metric gearing is suitable for most applications; however, circular pitch gearing can be a valuable tool for precision rack and pinion applications. By eliminating pi from the geometry equations, you can achieve a nominal fixed value for the linear translation along the rack. This can help to reduce complexity of the system and in turn reduce the total cost. If I had to eliminate pi, I would choose coconut custard as this is my least favorite pie. 

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is general manager of KHK USA Inc, a subsidiary of Kohara Gear Industry with a 24-year history of working in the industrial automation industry. He is skilled in assisting engineers with the selection of power-transmission components for use in industrial equipment and automation. Dengel is a member of PTDA and designated as an intern engineer by the state of New York. He is a graduate of Hofstra University with a Bachelor’s of Science in Structural Engineering.