Editing Spoke (section)

== Spoke length ==

When [[Wheelbuilding|building a bicycle wheel]], the spokes must have the correct length, otherwise there may not be enough threads engaged, producing a weaker wheel, or they may protrude through the rim and possibly puncture the inner tube.

=== Calculation ===

For bicycle spokes, the spoke length is defined from the flange seat to the thread tip. For spokes with bent ends, the nominal spoke length does ''not'' include the width of the spoke at the bent end.

For wheels with crossed spokes (which are the norm), the desired spoke length is
:<math>l = \sqrt{ d^2 + {r_1}^2 + {r_2}^2 - 2 \, r_1 r_2 \cos(a)} - r_3</math>
where
* '''''d''''' = distance from the center of hub (along the axis) to [[flange]], for example 30&nbsp;mm,
* '''''r'''''<sub>1</sub> = spoke hole circle radius of the hub, for example 19.5&nbsp;mm,
* '''''r'''''<sub>2</sub> = half of effective Rim Diameter (ERD), or the diameter the ends of the spokes make in a built wheel (see 'Discussion' attached to this article for explanation) of the rim, for example 301&nbsp;mm,
* '''''r'''''<sub>3</sub> = radius of spoke holes in the ''flange'', for example 1.1&nbsp;mm,
* '''''m''''' = number of spokes to be used for one side of the wheel, for example {{Sfrac|36|2}}=18,
* '''''k''''' = number of crossings per spoke, for example 3 and
* '''''a''''' = 360° ''k''/''m'', for example {{Sfrac|360° ⋅ 3|18}} = 60°.

Regarding '''''d''''': For a symmetric wheel such as a front wheel with no disc brake, this is half the distance between the flanges. For an asymmetric wheel such as a front wheel with [[disc brake]] or a rear wheel with chain [[derailleur]], the value of '''''d''''' is different for the left and right sides.

'''''a''''' is the angle between (1) the radius to a nipple hole in the rim to which a spoke is attached, and, (2) the radius to the flange hole holding the spoke. The spoke crosses either 1, 2, or 3 oppositely pointing spokes depending on the lacing design. On the flange, the angle between the radii of adjacent holes is 360°/''m'' (for equally spaced holes). For each spoke crossed, the hub is rotated with reference to the rim one "angle between adjacent flange holes". Thus, multiplying the "angle between adjacent flange holes" by ''k'' gives the angle '''''a'''''. For example, a 32 spoke wheel has 16 spokes per side, 360° divided by 16 equals 22.5°. Multiply 22.5° ("angle between adjacent flange holes") by the number of crossings to get the angle '''''a'''''—if 3-crosses, the 32 spoke wheel has an angle '''''a''''' of 67.5°. Regarding '''''r'''''<sub>3</sub>: The size of the spoke holes in the flange does not matter for the needed spoke length. This term removes the effect of the hole size. Since the holes are usually small (just over 2&nbsp;mm in diameter), the effect is small and in practice matters little.

For [[Wheelbuilding|radially]] spoked wheels (zero crossings), the formula simplifies to the [[Pythagorean theorem]], with spoke length '''''l''''' plus '''''r'''''<sub>3</sub> being the slope, '''''r'''''<sub>2</sub> minus '''''r'''''<sub>1</sub> being the base, and '''''d''''' being the rise:
:<math>(l + r_3)^2 = (r_2 - r_1)^2 + d^2</math>; or solving for the length
:<math>l = \sqrt{d^2 + (r_2 - r_1)^2} - r_3.</math>

[[Image:spoke-length.png|thumb|300px|right|A flat view of a crossed wheel with one spoke visible]]

=== Derivation ===

The spoke length formula computes the length of the [[space diagonal]] of an imaginary [[cuboid|rectangular box]]. Imagine holding a wheel in front of you such that a nipple is at the top. Look at the wheel from along the axis. The spoke through the top hole is now a diagonal of the imaginary box. The box has a depth of ''d'', a height of ''r''<sub>2</sub>-''r''<sub>1</sub>cos(''α'') and a width of ''r''<sub>1</sub>sin(''a'').

Equivalently, the [[law of cosines]] may be used to first compute the length of the spoke as projected on the wheel's plane (as illustrated in the diagram), followed by an application of the [[Pythagorean theorem]].