The Math Behind Involute Spur Gears

15 min.
Jan 20, 2024
Table of Contents

A Little History

Gears are fundamental to all systems where power transfer is important. Their inception dates to the ancient civilizations of Greece and China, where they were initially crafted in wood.

The Antikythera mechanism, an ancient Greek device featuring intricate gear systems, stands testament to the early mastery of gear technology. These marvels demonstrate an ingenious method of transmitting rotational force and controlling mechanical movement with precision. It contains advanced mechanical concepts that may resemble planetary gears, ring gears, gear sets and so on.

The Greek Antikythera Mechanism:


Image Credit: Encyclopædia Britannica, inc. (2023, October 13). Antikythera mechanism. Encyclopædia Britannica.


Image Credit: Tony Freeth/UCL
Fundamentally, gears are designed to alter the speed, torque, and direction of a power source. In their simplest form, they can be described as spinning levers. Gears are cylindrical or conical objects with teeth that mesh with the teeth of another gear. This interaction allows one gear, the driver, to turn another, the driven gear, thereby transmitting power efficiently and accurately.

The beauty of gears lies in their versatility. They are integral in countless applications, from the minute gears in watches to the massive gears in industrial machines. Gears ensure that clocks tick in unison, vehicles accelerate smoothly, and wind turbines harness energy efficiently. Their usage spans across various fields, including automotive, aerospace, robotics, and manufacturing.
This article focuses on the definition of spur gears. Other kinds of gears exist- bevel gears, which are good for transferring torque at 90°, helical gears which are spur gears whose teeth travel at a helical path, herringbone gears which have helical teeth that travel in both directions. All these gears are designed from the central principles of spur gears.

We can define spur gears and mathematically define them using the information below.

Important Terms

Pitch Circle

If you were to simulate gears as circles, the pitch circle would be the diameter that represents the pure rolling motion of the gear. Because teeth mesh, a gear would not accurately be modeled from the outer diameter of the teeth, nor the root diameter at the base of the teeth, but between the two as shown. In other words, it is an imaginary circle that passes through the point where the teeth mesh, and defines the true size of the gear. This is also referred to as pitch diameter or pitch circle diameter.

Addendum & Dedendum

The addendum is the radial distance from the pitch circle to the top of the tooth. This is particularly important relative to the pitch circle.

The dedendum is the radial distance from the base of a tooth to the pitch circle - again important and defined relative to the pitch circle.

Addendum Circle / Major Diameter

The circle whose diameter is made by adding the pitch circle and the addendum. This should equal the outer diameter of the gear. Interestingly, this is not commonly referred to as a major diameter.

Dedendum Circle / Root Circle

The root circle is a circle whose diameter is formed by the pitch circle subtracting the dedendum. This is also known as the dedendum circle, minor diameter, or root diameter.

Circular Pitch

The distance from the surface of one tooth to the same surface on the subsequent tooth. This is measured as an arc length at the pitch circle on the tooth’s surface, as shown below in red.

Diametral Pitch

This can be defined as a ratio of the number of teeth to the pitch circle. Measured in pitch per diameter size, almost always inches are used. For metric applications, module us usually used in place of diametral pitch. This is also known as circular pitch.

Diametric Pitch = Number of Teeth / Pitch Diameter. In this case, 15 / 1.5 = 10.

Involute Profile

This can be defined as part of a family of curves known as the roulette family. The curvature is represented by a point that “rolls” around another shape. For the specific involute curve, imagine wrapping a string around a cylinder, and then placing a pen at the end of the string. Then unwind the string from the cylinder as the pen draws the path. The path that is drawing is an involute curve.

Base Circle

The diameter of circle that marks the start of the involute profile.


For our dear friends who use the metric system, Module often takes the place of diametral pitch. Modulus is measured in millimeters per tooth. Please see below to compare the attributes of each.
Module and Diametral Pitch - Key Differences
Diametral Pitch Module
Unit of Measure: Inch mm
Implication of Size: Larger values indicate smaller teeth, more ‘teeth per inch’ Larger values indicate larger teeth, or more ‘mm per inch’
Geographic Usage: Mostly used in the United States and countries where the imperial system is in use Used in countries that use the metric system

The Mathematical Relationships

To design gears, we need only 3 values, arguably 4 if we wish to also define how thick the gear is. As an example, assume we want to have two gears work together, one gear having 15 teeth, the other having 20 teeth. The gear with 15 teeth will be designed here as an example.

For gears to be able to mesh together correctly, the must have the same pitch diameter or modulus, depending on what units you are working in.

Given the values for:

Diametral Pitch, P
Number of Teeth, N
Pressure Angle, θ (Degrees)

To solve for pitch diameter, or in other words, the diameter of the pitch circle, we use:
For our example, let’s say we would like to have our first gear with 15 teeth have a diametral pitch of 10. To calculate our pitch diameter:
It can be tempting to use the pitch diameter as an input and solve for the diametral pitch as it gives straightforward dimensional controls in defining the gear. However, since two gears need to have the same diametral pitch to mesh, it is usually best to start by defining the diametral pitch first.

From here, we can also solve for the base diameter, using the following equation:
The above equation asks for theta, or the pressure angle of the gear. The pressure angle on almost all gears is 20°. In less common cases, a pressure angle of 14.5° is used. Use 20° as a default, especially if you are unsure of what pressure angle to use. Using the pitch diameter of 1.5 that we found previously, and remembering that theta is expressed in degrees here, we find:
Addendum, a is simply the inverse of diametral pitch, or:
In our example, the gear would be:
The dedendum, denoted by b, would expressed as:
In our example, the gear would be solved as:
Addendum diameter can be found with the following:
In our example, addendum diameter would be:
Dedendum Diameter is given as:
In our example:
We can also calculate the degrees in which every tooth should be spaced. This is simply given in
degrees as:
Finally, the parametric curve equations to determine the curvature of the gear teeth. This is split into x values and y values, calculated as:
Where T values will run from 0 to 1. In our example, we have:
Plotting this parametric equation, we can visualize the following:
Interestingly, when this value is turned up to a maximum of 22, the parametric curve becomes clear:
And that math is all we need to define our gear.

We know where the roots of the teeth start, how to space them, what the outer boundary of the gear is, the profile of the tooth, and because we know the pitch circle diameter and diametral pitch of the gear, we can define how far apart each gear should be from the other and that the teeth will fit with the other gear teeth properly. Even better, if you negate how thick you want the gear to be, we have been able to define all this with only 3 numeric inputs.

Automating Gears in Alibre Design

We could enter all this into our CAD program to start defining gears, but in Alibre Design, the work has been done for you. To access gear scripting, go to the Scripts tab and launch the python console.
In the console, select Examples and then mechanical.
You can use the gear generation script that comes in Alibre, import your own, or write one from scratch!

Here, using an imported Gear Generator (DP) script, we are able to define and generate a gear model from the three parameters we used in the example.
And here, we are able to see a solid model of the gear we defined mathematically, and each gear tooth has a perfect curvature.
Alibre makes defining gears convenient, because having the ability to use solid models makes it easy to instantly generate gears, measure the diameter of circles, measure the distance of spur gear features or between spur gears, increase or decrease the distance between gears if needed, or make just about any updates. It works as a perfect pitch diameter calculator. Now you’re qualified to design all kinds of gear teeth, except perhaps for the 32 teeth in humans or their wisdom teeth!

Formula Summary

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