Bevel and Hypoid Gear Geometry




This document specifies the geometry of bevel gears. The term bevel gears is used to mean straight, spiral, zerol bevel and hypoid gear designs. If the text pertains to one or more, but not all, of these, the specific forms are identified. The manufacturing process of forming the desired tooth form is not intended to imply any specific process, but rather to be general in nature and applicable to all methods of manufacture. The geometry for the calculation of factors used in bevel gear rating, such as ISO 10300 (all parts), is also included. This document is intended for use by an experienced gear designer capable of selecting reasonable values for the factors based on his/her knowledge and background. It is not intended for use by the engineering public at large. Annex A provides a structure for the calculation of the methods provided in this document.


For many decades, information on bevel, and especially hypoid, gear geometry has been developed and published by the gear machine manufacturers. It is clear that the specific formulae for their respective geometries were developed for the mechanical generation methods of their particular machines and tools. In many cases, these formulae could not be used in general for all bevel gear types. This situation changed with the introduction of universal, multi-axis, CNC-machines, which in principle are able to produce nearly all types of gearing. The manufacturers were, therefore, asked to provide CNC programs for the geometries of different bevel gear generation methods on their machines.

This document integrates straight bevel gears and the three major design generation methods for spiral bevel gears into one complete set of formulae. In only a few places do specific formulae for each method have to be applied. The structure of the formulae is such that they can be programmed directly, allowing the user to compare the different designs.

The formulae of the three methods are developed for the general case of hypoid gears and to calculate the specific case of spiral bevel gears by entering zero for the hypoid offset. Additionally, the geometries correspond such that each gear set consists of a generated or non-generated wheel without offset and a pinion which is generated and provided with the total hypoid offset.

An additional objective of this document is that, on the basis of the combined bevel gear geometries, an ISO hypoid gear rating system can be established in the future.

ISO 23509:2016 was developed by Technical Committee ISO TC 60, Gears. The changes in the new document include:

  • minor corrections of several formulae;
  • the figures have been reworked;
  • explanations have been added in 4.4;
  • the structure of Formula (129) has been changed to cover the case zm = 0°;
  • a formula for the calculation of cbe2 has been added as Formula (F.160);
  • the values for αnC and αnD in Formulae (F.318) and (F.319) have been extended to three decimal digits to prevent rounding errors.

The AGMA Bevel Gearing Committee approved the adoption of ISO 23509:2016 in June 2017. It was approved as an American National Standard, ANSI/AGMA ISO 23509-B17, on December 12, 2017.

ANSI/AGMA ISO 23509-B17 replaces ANSI/AGMA ISO 23509-A08. The new document represents an identical adoption of ISO 23509:2016, which replaced ISO 23509:2006.

Identical to ISO 23509:2016

ISBN: 978-1-64353-002-4

Pages: 149

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