Bike Spoke Length Calculator

Sit on a bicycle and the wheels do not push back the way a table leg does. Each one is a thin hoop tied to a hub by a few dozen wires about two millimeters thick, and not one of those wires could hold a rider up if you stood it on end. The wheel works because every wire is already pulling. It keeps working only if every wire is the right length.

That second condition sounds dull. It is also the one that decides whether the first can ever be met at all.

Carrying a load by letting go

The first bicycle wheels were built like cartwheels, with stout wooden spokes squeezed between hub and rim. Matthew Ford’s doctoral study of the wheel at Northwestern University dates the reversal to 1869, when the French mechanic Eugene Meyer patented a wheel whose slender iron spokes were each pulled tight against a rim held in compression. James Starley’s tangent-spoked wheel followed in 1874, and Ford notes that the principle has not changed much since. The trick, as he describes it, is that a spoke under tension can support compression by losing some of its pull, though never all of it.

So the spokes that do the most work are the ones at the bottom. As Park Tool explains, a wheel flattens slightly where it meets the ground, and the tension of the spokes there drops. The cycle repeats on every turn. Jobst Brandt made the same point in fewer words: in normal use, a wheel only unloads spokes. Two engineers who tested a strain-gauged wheel, in a paper published in 1993, found that a pneumatic tire reduces the peak stress changes in the spokes.

This is why tension has to sit in a band. Too little, Park Tool warns, and the spokes relax too far on each turn, which stresses spoke and rim alike and lets the wheel lose tension and drift out of true. Too much, and rims crack or spokes break. Most rims, it says, carry a suggested range of 100 to 120 kilograms-force. Brandt put the usual figure at no more than a third of the wire’s yield stress.

What tension does not much care about is the pattern. Henri Gavin fitted strain gauges to three rear wheels laced two-cross, three-cross and four-cross, then rode them. Under the rider’s weight, the strain in a spoke was nearly the same in all three. Lacing is chosen for other reasons.

United States consumer rules treat the result as a single structure. A bicycle wheel may have no missing spokes, and its rim must keep hold of spokes and tire while 2000 N presses sideways on the axle for at least thirty seconds. If the hub is offset, the regulation says to push in the direction of the offset. Hold that detail for a moment.

Four measurements and one triangle

Look at a wheel from the side. A spoke, the line from the axle to its hole in the hub flange, and the line from the axle to its hole in the rim form a triangle. Two sides are known, half the flange diameter and half the rim diameter, and the angle between them is fixed by the lacing, so the cosine rule gives the third.

Now look from behind. The flange sits to one side of the rim, which makes that third side the base of a right triangle, and Pythagoras finishes the job. A small correction for the width of the hole in the flange comes off the end. John Allen sets the steps out on Sheldon Brown’s site, and the hub dynamo maker Schmidt prints the same square root in its instruction sheet.

The rim number matters most, and it is the least obvious. Effective rim diameter, ERD, is not really a property of the rim. Damon Rinard defines it as the diameter on which you want the ends of the spokes to lie, and says most people prefer that to be near the end of the nipple. Park Tool says a better name would be effective spoke end diameter.

Park Tool’s way of measuring it needs two spokes of known length. Thread each into a nipple through opposite holes until the spoke reaches the bottom of the screwdriver slot, pull the two toward each other, and measure the gap between the elbows with a caliper. In the company’s worked example the spokes are 266 mm each and the gap is 35.7 mm, so the ERD is 567.7 mm. Not everyone stops at the same point. Rohloff’s handbook subtracts the height of the nipple head twice, while Allen adds about 4 mm to a measurement taken on an empty rim. A published ERD is useful only if you know which convention produced it.

The hub is more forgiving. Flange diameter is measured between the centers of opposite spoke holes, not across the outside of the flange, and Rinard puts it between 38 and 67 mm on most hubs. Flange distance is half the width over the locknuts minus the distance from locknut to flange. Allen’s table of errors is reassuring here: a mistake in rim diameter passes half its size to the spoke, while 1 mm of error in flange distance costs about a tenth of a millimeter.

Crosses, torque and the dish

The angle in the triangle comes from the pattern. Divide the full circle by the number of spokes on one flange and multiply by the number of crosses. For 32 spokes laced three-cross, that is 360 divided by 16, times 3, or 67.5 degrees. Set the crosses to zero and the triangle collapses into a straight line from hub to rim, which is radial lacing.

More crosses make a longer spoke that leaves the flange closer to a tangent. Sheldon Brown’s guide says that is what lets a wheel resist the twisting of hard pedaling or of a hub brake, and that radial lacing suits only front wheels without one. The United Bicycle Institute teaches the same rule. Makers add limits of their own. Schmidt does not permit radial spoking of its dynamo hub. Rohloff, whose gear hub has a hole circle 100 mm across, requires two crosses in large wheels and one in small ones, never three.

Then there is the asymmetry the federal test allows for. Park Tool states the principle simply: the rim is centered between the faces of the hub that touch the frame, wherever the flanges happen to sit. On a rear hub the right flange is pushed inward to make room for the sprockets. Allen gives typical distances of 35 mm on the left and 20 mm on the right. The wheel is dished, like a shallow bowl, and its right spokes stand nearly upright.

Take the rear wheel Park Tool measures. It has 32 holes, an ERD of 567.7 mm, and 61 mm flanges set 32.5 mm and 18.7 mm from center. Spoke holes are 2.5 mm, and the rim is drilled 4.3 mm off center. Suppose it is laced three-cross with the offset toward the left. That moves 4.3 mm from the left flange distance to the right, giving 28.2 mm and 23.0 mm. The cosine rule puts the side-view length at 273.63 mm. Pythagoras and the hole correction give 273.8 mm on the left and 273.3 mm on the right, the same figures the United Bicycle Institute’s calculator returns. You would buy 274 mm and 273 mm.

Why the tight side stays tight

The rim sits still only if the sideways pulls of the two flanges cancel. Each spoke’s sideways pull is its tension multiplied by the sine of its bracing angle, the angle between the spoke and the plane of the rim. A steep spoke pulls sideways weakly, so it must be tighter to hold its own. Ford’s analysis says exactly this, that the spokes on the steep side must be tighter by the ratio of the sideways lean, and he puts the factor near 1.5 for a typical wheel. Gavin notes that the right-side spokes can need up to three times the pretension of the left.

In the example the bracing angles are 5.9 degrees on the left and 4.8 on the right, so the left spokes carry 82 percent of the right-side tension. Set the rim offset to zero and the angles become 6.8 and 3.9 degrees. The left side falls to 58 percent. That gap is what an off-center rim buys, and it is why Park Tool and Rinard both describe such rims as a way to even out tension.

Different spoke lengths do not do the same thing. Park Tool is explicit: a separate length per side serves thread engagement and does not equalize tension. Nor is the imbalance a fault. The target simply applies to the tighter side.

Length matters at the scale of threads. Spokes are usually sold in steps of 1 mm, sometimes 2. Park Tool allows a range of one to two millimeters; Allen says about three, and shrinking as short nipples come into fashion. A spoke that is too short leaves threads showing and stops short of the nipple’s head, which can then break, especially in aluminum. One that is too long pokes toward the tube or runs out of thread before the wheel is tight. Even the authorities split over rounding. Brown rounded up, while Allen and Schmidt both lean a millimeter shorter.

What the triangle leaves out

The formula is geometry and nothing more. A properly tensioned spoke stretches by about 0.3 percent, in Allen’s estimate, and the rim shrinks a little, which together make a spoke of 250 to 300 mm behave about 1 mm longer than calculated. Nipple length moves the point where threads run out. The triangle also assumes spokes with an elbow, in evenly spaced holes, in equal numbers on each side. Straight-pull hubs and paired holes need other methods, and Park Tool’s own procedure sets straight-pull spokes aside as well.

The tension figure is narrower still. It describes a centered rim at rest, with every spoke on a side pulling equally. Real wheels scatter, and Park Tool accepts a spread of 20 percent either side of the average.

Allen opens his article with a line he credits to Howard Sutherland: one measurement is worth 50 expert opinions. He also warns that you learn whether the length is right only once the wheel is mostly laced. Both remarks point the same way. Measure the rim and hub in your hands, run the numbers, and try a few spokes before trusting the rest.