Geometry of involute gears | What is an involute | module | pitch circle | simply explained

Involute gearing plays a central role in mechanical engineering due to its efficient power transmission in gear systems. The tooth flank shape is created by unrolling a straight line on a base circle, forming what is known as the involute. This special shape leads to favorable meshing conditions and is both easy and cost-effective to produce, which is why it has become the standard. The involute can be visualized by imagining a taut string or a straightedge being rolled without slipping around a circle. The resulting curve describes the tooth flank. The further the involute extends, the larger its radius of curvature becomes, resulting in a flatter tooth flank. This property is used in profile-shifted gears to reduce load and wear. A gear is characterized by various diameters: base circle, pitch circle, addendum circle, and root circle. The pitch circle serves as a reference on which tooth thickness and gap are measured. The addendum circle marks the top of the tooth, the root circle its base. There is no force transmission within the base circle, as the involute does not exist in that region. A crucial factor for gear compatibility is the module, which defines the tooth size. Only gears with the same module can mesh correctly. The module is directly related to the circular pitch (the distance between two similar flanks on the pitch circle) and the pitch diameter, which is calculated as the product of module and number of teeth. For non-profile-shifted gears, tooth thickness and gap are each half the circular pitch, resulting in a backlash-free pairing that operates quietly and without play. The addendum and root circle diameters can also be calculated from the module and pitch diameter, with root diameter additionally factoring in clearance to avoid interference between meshing teeth. The standard center distance describes the ideal spacing between the axes of two meshing gears without profile shift. At this distance, the rolling circles (pitch circles) are identical. When the center distance changes, the rolling circles also change, while the pitch circle remains constant. An important factor in the tooth shape is the standard pressure angle. Although all flank shapes are involute, their appearance varies depending on the base circle diameter used in their construction. Smaller base circles result in sharper teeth, larger ones in blunter teeth. To ensure that matching flank shapes are created despite different gear sizes, the base and pitch diameters must maintain a constant ratio. This ratio is defined by the standard pressure angle, which is typically 20° in practice, though other values such as 14.5° or 25° are also used. Gears are often manufactured by hobbing, where the inclination of the cutting edges in the rack-shaped tool corresponds directly to the pressure angle. This makes production easier and more cost-effective, as no complex tool geometries are required—only straight cutting edges. To summarize: the pressure angle determines the tooth flank shape, the module defines the size of an individual tooth, and the pitch diameter defines the overall gear size. Tooth spacing can be referenced not only on the pitch circle but also on the base circle, in which case it's called base pitch or line of action pitch. This constant base pitch is a direct result of the involute shape and ensures smooth, uninterrupted power transmission during gear engagement. 00:00 Use of involute gears 00:24 Constructing an involute (unwinding a thread) 01:23 Constructing an involute (rolling a straight line) 02:03 Radius of curvature 03:34 Nomenclature 05:04 Standard reference pitch circle 06:04 Tooth size: the module 07:13 Gear size: the standard reference pitch diameter 08:08 Circular pitch 09:00 Diametral pitch 10:08 Circular tooth thickness & tooth space width 10:49 Tip circle diameter & tooth root diameter 11:23 Standard center distance 12:17 Operating & reference pitch circle (difference) 13:45 Tooth shape: the pressure angle 15:04 Geometric similarity of involutes 17:02 Tooth shape: standard pressure angle 18:50 Gear cutting by hobbing 20:20 Base pitch (meshing pitch)