Looking for trig diagrams

Hello, I am looking for diagrams or links to diagrams of the calculations for measuring threads over wires.

here are the specific formulas in question:

major diameter D = largest diameter

pitch p = 1 / (threads per inch)

best wire size W = .55735 * p

measurement over wires M = D – (0.866025*p + 3W)

as per the machinist handbook (#28)



Specifically, the magic numbers .55735 and .866025 drive me crazy, and I never remember these, but if I can derive them from a picture, then remembering the exact formula is easier: eg. 12/pi is much easier to remember than 3.81972, and i can extend 12/pi to any precision desired.

the magic numbers fundamentally limit how accurate parts can be made if they are rounded numbers.

thanks!