The geometry, in one sentence
A spoke runs from a point on the hub flange to a point on the rim, and those two points sit on circles of different radii that are separated along the axle. Put the hub centre at the origin and the length is the straight-line distance between them, which is a three-dimensional Pythagoras problem: the radial separation, the angular separation around the wheel, and the sideways offset of the flange.
Written out, with R as half the ERD, r as half the flange diameter, O as the flange offset from the centreline and a as the angle the spoke sweeps around the hub:
length = sqrt(R² + r² + O² − 2·R·r·cos a) − hole/2
The angle a is where the cross pattern enters. Each spoke on one flange is separated from its neighbour by 360 degrees divided by the spokes on that side, and a spoke laced to cross X others sweeps X of those gaps in each of two directions, which works out to a = 720 × X / total spoke count. For a 32 hole wheel laced three cross that is 67.5 degrees. Radial lacing sets X to zero and a to zero, which collapses the formula to the plain right triangle you would expect.
The final subtraction accounts for the spoke bending around the edge of its hole in the flange rather than pivoting at the hole centre. Half the hole diameter is the conventional allowance, and 2.6 mm holes are common, so 1.3 mm comes off.
Why the ERD is the number that ruins builds
Effective rim diameter is the diameter of the circle through the points where the spoke nipples seat. It is not the bead seat diameter, it is not the outside of the rim, and it is not a number you can take off a tape measure laid across the rim. On a deep section rim the gap between what you can see and what the spoke actually reaches is large.
Manufacturer figures exist and are frequently wrong, sometimes by two or three millimetres, which is exactly the margin that turns a good build into a bottomed-out nipple. Rims also change between production runs without the published figure being updated. The reliable method is to thread two spokes into two nipples at opposite holes, join them with a piece of threaded rod or a spoke with the head cut off, measure the assembly, and add the length of the two spokes back. Do it twice at ninety degrees apart and average.
One more definitional trap: some builders quote ERD to the top of the nipple and some to the seat inside it. A two millimetre difference in convention produces a two millimetre error in every spoke. Whatever definition your measuring method produces, use the same one consistently and check a single spoke against a trial fit before ordering thirty-two of them.
Dish, bracing angle and why the left side goes slack
On a rear wheel the cassette occupies space on the drive side, so the drive flange sits closer to the centreline than the other one. The rim still has to end up centred between the dropouts, so the shallower drive-side spokes must pull harder to hold it there. The result is a tension split that can leave the non-drive side at half the drive side tension or less.
| Wheel | Typical offset split | Consequence |
|---|---|---|
| Front, rim brake | Near symmetrical | Even tension, the most durable case |
| Front, disc brake | Mildly asymmetric | Brake side carries more, still comfortable |
| Rear, rim brake | Strongly asymmetric | Non-drive spokes loosen and fatigue first |
| Rear, disc and wide cassette | Most asymmetric | The classic case for an asymmetric rim to claw some back |
The bracing angle figure in the results is the angle each side makes against the wheel plane. It is what actually resists sideways loads, and a wheel with a poor bracing angle on one side feels vague under load however tight the spokes are. The tension ratio shown alongside it is the reason rear wheels fail on the side that was never doing much work: low tension lets the spoke go slack under load, and a spoke that cycles between slack and loaded fatigues at the elbow.
Cross pattern, in practical terms
Three cross is the default for 32 and 36 hole wheels because it gives a spoke angle close to tangential, which is what transmits drive and brake torque efficiently, while keeping the spoke length sensible. Lower counts force lower crosses simply because there are not enough spokes to cross three of them: a 24 hole wheel with 12 spokes a side runs out at two cross, and 16 hole wheels are typically one or two.
Radial lacing appears on front non-disc wheels, where there is no torque to transmit. It gives the shortest spokes and the stiffest-feeling wheel laterally, and it puts the whole load straight down the spoke. It is a poor choice anywhere torque is applied, and it loads the hub flange in a direction some hubs are explicitly not made for. That is a question for the hub, not for the calculator.
What this cannot do for you
It computes a length from dimensions. It does not know whether your spokes are the right gauge for the load, whether the rim is rated for the tension your build needs, or whether the hub permits the pattern you chose. It does not know about asymmetric rims, where the spoke bed is offset and the effective radius differs by side, and it does not model the small length change that occurs as the wheel comes up to tension.
Order one spoke first if the cost of being wrong is thirty-two spokes. If the wheel is going onto a bike you depend on, the rest of the mechanical picture is worth a pass through the bike maintenance guide, and once it is built and rolling the drivetrain side is covered by the chain length and wear calculator.
Questions people ask
How do I measure ERD without a proper tool?
Thread a nipple onto each of two spokes so the spoke end sits exactly where a spoke end would sit in a finished wheel, drop them into holes directly opposite each other, and join them across the rim with a straightedge or a length of threaded rod. Measure the gap between the two spoke ends, then add the two spoke lengths. That total is the ERD under the convention that matters, because it is measured to the same surface the spoke will actually reach. Repeat at ninety degrees and average, since rims are not perfectly round off the shelf. If the two readings differ by more than about half a millimetre, check for a bent section before building.
What happens if the spokes are one millimetre off?
One millimetre is usually recoverable and two usually is not. Too long, and the spoke end protrudes past the top of the nipple and can push into the rim tape and the tube; on a tubeless setup it can compromise the seal. Too short, and the threads engage too little of the nipple, so the nipple strips or the spoke pulls out under tension, and you may be unable to reach target tension at all. Since spokes are commonly stocked in 2 mm steps, being one millimetre off is normal and builders round toward the shorter side on the drive side of a dished rear wheel, where the nipples run further onto the thread anyway.
Can I use the same length for both sides of a rear wheel?
Only if the calculation says so, and on a dished rear wheel it usually does not. The offset difference between drive and non-drive flanges is typically 15 mm or more, and the flange diameters often differ too, which together commonly produce a length difference of two to four millimetres. That is more than a rounding step, so ordering one length for both sides means one side is wrong. Some hub and rim combinations do land close enough to share a length after rounding, which the difference figure in the results tells you directly.
Does spoke tension change the length I need?
It changes the built length by a fraction of a millimetre, which is well inside the rounding you are already doing. A spoke stretches elastically under tension, and the rim compresses slightly, but both effects are small next to a 2 mm stocking step. What does matter is that the calculation assumes the spoke runs in a straight line from the flange hole to the nipple seat. That holds well for laced patterns; it holds less well for very low spoke counts with extreme angles, where builders sometimes find the real requirement sits half a millimetre off the computed figure.
Why does my calculated length differ from the shop calculator I used?
Almost always because of a different convention rather than different arithmetic. The three usual culprits are the head allowance, where some calculators subtract half the spoke hole diameter and some subtract nothing; the ERD definition, measured to the nipple seat in some places and to the top of the nipple in others; and the flange offset, which some sources give as centre-to-flange and others as flange-to-flange or as a locknut-referenced dimension. Check those three against whatever you are comparing to before assuming one of them is wrong, and if the difference is exactly 1.3 mm you have found the head allowance.