What the inputs mean
- ERD (effective rim diameter): the diameter at the spoke ends when they sit in the nipples. Rim makers publish it; you can also measure it.
- Flange diameter: the diameter of the circle through the centers of the spoke holes on a hub flange, not the outside of the flange.
- Center to flange: the distance from the middle of the hub (halfway between the locknuts) to the middle of that flange. On a rear hub or a disc front hub the two sides differ.
- Spoke hole diameter: the hole in the flange. Half of it comes off the length, because the spoke bends through the hole.
- Spokes and crosses: the total count and how many other spokes each one crosses on its side (0 is radial).
The formula
α = 720 × crosses ÷ spokes (degrees)
L = √(d² + r1² + r2² − 2 × r1 × r2 × cos α) − hole ÷ 2d is center to flange, r1 is half the flange diameter and r2 is half the ERD. The square root is the straight line from the flange hole to the spoke end: the r1, r2 and cos α part is the law of cosines in the wheel's plane, and d adds the sideways offset.
Worked example
Example values, not from any product: ERD 600 mm, 32 spokes, 3 cross, 2.6 mm holes, both flanges 45 mm, center to flange 35 mm on the left and 20 mm on the right.
- α = 720 × 3 ÷ 32 = 67.5°, r1 = 22.5, r2 = 300.
- Left: 35² + 22.5² + 300² = 91,731.25. 2 × 22.5 × 300 × cos 67.5° = 5,166.23. √(91,731.25 − 5,166.23) = 294.22, minus 1.3 = 292.9 mm.
- Right, with 20 mm: 291.5 mm.
The side with the flange closer to the center (here the right) takes the shorter spoke. Changing to 2 cross drops the left spoke to 285.4 mm; radial (0 cross) gives 278.4 mm. A 2 mm error in ERD moves the result by about 1 mm (293.9 mm at ERD 602).
Before you order spokes
The formula is only as good as the measurements. Use the numbers the rim and hub makers publish for your exact parts, or measure them, and check how your nipples and spokes are meant to sit. How you round to available lengths is your call as the builder. For the rim sizes behind ERDs, see the bike tire size chart.
Questions
How do you calculate spoke length?
L = √(d² + r1² + r2² − 2 r1 r2 cos α) − hole ÷ 2, with α = 720 × crosses ÷ spokes. For ERD 600, 45 mm flange, 35 mm center to flange, 32 spokes 3 cross and 2.6 mm holes, that is 292.9 mm.
Why are left and right spokes different lengths?
The flanges usually sit at different distances from the hub center, and sometimes have different diameters. In the example, 35 mm and 20 mm give 292.9 mm and 291.5 mm.
What does 3 cross mean?
Each spoke crosses three others on the same side between hub and rim. With 32 spokes that sets the angle α to 720 × 3 ÷ 32 = 67.5°.
What is ERD on a rim?
Effective rim diameter: the diameter where the spoke ends sit inside the rim. It is not the same number as the bead seat diameter; take it from the rim maker or measure it.