It's a complicated enough formula with core characteristics,
AC voltage and Frequency.
I have only designed some switching toroids, but the
general formula for that is:
E = (4.4 x 10^-8) * F * N * B * A
E is the peak voltage
F is the frequency
N is the number of turns in the winding
B is the core flux density
A is the cross section of the core
A number of books deal with this issue; I pulled this out
of Gottlieb's "Power Supplies - Switching Regulators and
Converter's" which is not necessarily a highly rated book
according to the reader feedback on Amazon, and there
are probably better.
At the web site www.toroid.com/kits.htm, you can find toroidal transformers kits from 80VA-750VA (I think). And with each kit the manual is suppose to have info and pictures of how to wound and calculate the primary and secondary voltages. The company is Toriod Corporation of Maryland which is pretty well know.
Also at SDS Labs Audio Design there WAS a how to section with info and pictures of how some one built up one of these toroid kits. The SDS Labs web site can be found in the Directory section below on DIYaudio site under Amplifiers in Projects.
I mean WAS cause I have tired to enter the site but I get" Page Not Found" message. Maybe they are updating the site or something.
I don't know the availability because I have a 1980 edition, but there is a Howard Sams book on transformers.
"Practical Transformer Design Handbook" by Eric Lowdon
First Edition, First Printing - 1980
Howard W. Sams & Co., Inc.
It's a 240pg soft cover book, and is 8-1/2" X 11". It starts at "Elementary Electromagnetics" (Chapter 2) and covers through properties, construction, design, applications, examples, and practical issues like buying new or used, and Tests and Measurements (Chapter 16).
One drawback is that it only has a few pages on toroids. One quote from the book is:
...anyone who proposes to wind large numbers of turns onto a toroid is subjecting himself to cruel and unusual punishment.