Spoke — bicycle spoke-length calculator

The theory

A spoke runs from a hole in the hub flange out to a hole in the rim. Because the flange is offset sideways from the rim's plane and the spoke is angled tangentially to cross its neighbours, the spoke is the space-diagonal of a small 3D triangle. Its length is:

L = \sqrt{d^{2} + R^{2} + r^{2} - 2 R r \cos θ} - h

d — distance from the hub centre to the flange.
R — effective rim radius (ERD ÷ 2).
r — radius of the spoke-hole circle in the flange.
h — radius of the spoke hole in the hub flange.
θ — the crossing angle, θ = 4πk / N, for k crossings and N spokes.

The square-root term is the straight-line distance between the two holes, derived from the law of cosines applied in three dimensions: the tangential angle θ separates the flange and rim holes around the axle, while d accounts for the sideways flange offset. Subtracting h gives the length measured to where the spoke actually seats in the flange. For a radial build (k = 0) the formula collapses to √(d² + (R − r)²) — simple Pythagoras.