If I understand Your question, there is no direct relationship between the face width and tooth height of gear. But all gear parameters determine load capacity of gear . Perhaps you will be able to read my article TO THE PROBLEM OF THE INVOLUTES GEAR PARAMETERS OPTIMIZATION.

@Alexander Petrovskiy: Thank you sir for your suggestions but i am not able to find your paper which you have suggested if possible could you please provide me the link of that paper.

The link of that paper: https://www.researchgate.net/publication/260436829_TO_THE_PROBLEM_OF_THE_INVOLUTES_GEAR_PARAMETERS_OPTIMIZATION_%28_____%29?ev=prf_pub

Article TO THE PROBLEM OF THE INVOLUTES GEAR PARAMETERS OPTIMIZATION...

Thank you sir for sending the link i am afraid to say that i cant understand this paper as it is written in Russian language but thanks for your kind suggestions.

This translator from Russian into English;

https://translate.google.com/#ru/en/%D0%AD%D1%82%D0%BE%20%D0%BF%D0%B5%D1%80%D0%B5%D0%B2%D0%BE%D0%B4%D1%87%D0%B8%D0%BA%20%D1%81%20%D1%80%D1%83%D1%81%D1%81%D0%BA%D0%BE%D0%B3%D0%BE%20%D0%BD%D0%B0%20%D0%B0%D0%BD%D0%B3%D0%BB%D0%B8%D0%B9%D1%81%D0%BA%D0%B8%D0%B9

Thank you sir for your kind support :)

it depends strongly on the shape of the gear and from the shift coefficient

Gear width is not affecting tooth height. Face width basically is determined by the available space in the transmission and the load capacity.

Thank you sir

Yes. Actually, both of them are related to the gear module. According to "Machine Design" book in "gear" section, the gear face width can be between 3*pi*module to 5*pi*module (approximately 9.4*module to 15.7*module). On the other hand it is best known that in the standard gears, the tooth height is equal to 2.25*module.

Thank you sir, if possible please provide me the details of the book you are citing.

These equations you refer is ''a theoretical guide'' to know how much approximately you can consider your face width, but in real transmission design you are not obliged to use exact result from equations. You need sometimes to increase the face width to reduce the stresses (bending and contact) or to increasing the Contact ratio for improving NVH. on the other hand the available space is limit. As a result you can change the face width with keeping the same tooth height and other parameters.

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition

III. Design of Mechanical Elements

14. Spur and Helical Gears

page 717 (or 719) below the equation (14-8) :

"As a general rule, spur gears should have a face width F from 3 to 5 times the circular pitch p."

As upper mentioned, there is not a direct relation between face width and tooth height. As I have an experience of over 25 years in the design of gears, I would be happy to assist you in any problem related to gears.

@Zoltan Korka: Thank you sir for your kind support.