How do I calculate spoke length?

Question

I have lots of bikes and lots of interesting wheel sizes, I would like to know how to calculate the right spoke lengths for a wheel.

I appreciate that there are probably rather a lot of variables but this is something that has been an issue for me from time to time so being able to work out the right spoke length for an arbitrary wheel (did I mention that I have half a dozen or more different wheel sizes in the garage?) would be useful.

To be clear - I don't want a calculator, I want to know what the sums are.

Answer 1

The basic arrangement for cross-laced spokes is thus:

Spoke Length = sqrt[ (RRSP - (HSR * cos(SAA)))^2 + HFO^2 - (HSR * cos(SAA))^2 ]

For radial (straight) spokes, the formula is simpler:

Spoke Length = sqrt[ (RRSP - HSR)^2 + HFO^2 ]

• RRSP (Rim Radius plus Spoke Penetration) is half of the Effective Rim Diameter given by the manufacturer plus 2mm for the spoke's penetration into the nipple
• HSR (Hub Spoke Radius) is the radius from the hub center to the outermost edge of the spoke holes on the hub
• SAA (Spoke Anchor Angle) is the angle of from the hub spoke hole to rim hole that the spoke goes to. It depends upon the lacing pattern and which individual spoke you are calculating for.
• HFO (Hub Flange Offset) is the distance from the hub's flange to the lateral center of the wheel, based upon the dish of the wheel. For a front wheel, this will be the same on each side, but for a rear wheel, which has a dish to accommodate the rear gears, this is different for each side since the hub's own lateral center is different from the wheel's lateral center.

I'll credit this page for the formula; it also has more detail.

There are also a bunch of calculators online, the easiest of which is probably Damon Rinard's spocalc spreadsheet. Other calculators:

• Danny Epstein's
• Roger Musson's (he literally wrote the book on wheel building)
• United Bicycle Institute

As usual, Sheldon Brown has a great page on wheel building and you can learn more about the math from the above mentioned book.

Answer 2

If you're still interested, you can check out the site I'm developing that draws the wheel for you, allows you to play with all the variables and select different lacing patterns. It may help to visualize what you want before you build it (and even see it spinning).

Note: It's still in active development, so don't expect perfection.

Answer 3

The wheel illustration above would appear to be a non-drive side radial lace and a drive side 3cross rear wheel. I've used this combo with a full radial laced front on a 24" BMX Cruiser I built in the late 90's. I referred to the rim manufactures data base through a distributer and found their recommendation to be spot on. I have found some spoke lengths can be hard to find. A Phil Wood Spoke Cutter/Threader is a vital tool to have access to in situations like that.