Double-butted when you spoke? Part III

Please bear with me for another post on bicycle wheels, and the math that goes along with them.

The most mathematically-complex issue in building bike wheels is getting the spoke length right. Why? How is it done? What are the options?

1. You can figure out which spoke you need by trial and error, IF you have a large number of different-sized spokes and you know what you are doing, and you don't mind taking a lot of time to build and rebuild a wheel until you get it right.
2. You can use moderately-long spokes and chop off the ends that would otherwise perforate your tire. Then take one out and measure it. There's your length.
3. You can use a spoke length calculator. Some of them require lots of precise measurements of your components but they do the math for you. Here's one calculator and here's another one. 
4. You can use a book that has lengths already recorded in chart form. This works IF you use components that were measured by the author of the chart.
5. You can do the math yourself. But not with what you learn in Excel Math. The calculation is a bit harder-than-average 6th grade math. 

Here's a diagram of a rear hub and the dimension you measure on a hub. Overall width across the locknuts (where the fork fits), the width between the flanges, and the widths of either side outside the flanges. You measure the same things on front hubs; they are less complex because they don't have gearing.

For some charts you need the effective rim diameter (ERD) which is from side to side inside the rim where the spoke enters the nipple. Plus the length of 2 nipples. ERD =  A + (B x 2). Other methods use the individual measurements rather than ERD>

And then add some geometry and math. This isn't a perfect diagram but it shows where the spoke fits  between the rim and the hub. We are calculating the length of an angled line between points on two circles. With a tolerance of a millimeter or two.

I was going to spare you the actual formula, but what the heck. It's not too long, too complex, and too messy, is it?

Yes, it is too long and messy. And once you get a result then you still have to build a wheel!

Making good wheels is difficult. It's very much a "craft" with some science mixed in. You can try it at home, but be forewarned, it's like rocket science. It's safer to send your wheel needs to an expert.