Finding Spoke Length

Yes, there is math involved.

First, a trigonometry refresher:

SOH CAH TOA:

Sin(θ) = Opposite ÷ Hypotenuse

Cos(θ) = Adjacent ÷ Hypotenuse

Tan(θ) = Opposite ÷ Adjacent

Pythagorean Theorem:

a² + b² = c²

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θ

Adjacent

Opposite

Hypotenuse

a

b

c

Notes about this example

• Finding spoke length is a three dimensional problem, but we can break it up into two two dimensional problems. First the side view, then the front view. In the case of radially laced spokes you can skip straight to the front view.
• This example will be using several small equations and we will be introducing compounding rounding errors. Either use this information to derive one big equation or use more decimal places for more accuracy.
• There is no allowance for spoke elongation, but it is generally okay to disregard unless working with exceptionally thin spokes.
• Our example will use 36 hole drilling and 3 cross lacing

Measure Hub & Rim (we are only calculating one side in this lesson)

℄

21mm

45mm

605mm

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

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2.6mm

Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

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We are working from a reference point of the spoke hole at the bottom of the rim. The hole in the hub is chosen simply by counting over 3 holes for 3-cross.

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We can solve this up by breaking the problem up into triangles.

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Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

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?

?

?

Okay, we don’t know any of the sides of this triangle, but we can draw another to help us out.

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

​

Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

​

?

?

?

We know the hub holes are half the PCD from the center, 22.5mm. The angle between each spoke hole can be found by dividing 360° by the 18 holes on this side of the hub, 20°.

22.5mm

60°

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

​

Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

​

?

?

19.49mm

Now let’s find the length of the two other sides of this helper triangle:

Sin(60)*22.5=19.49

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Cos(60)*22.5=11.25

22.5mm

60°

11.25mm

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

​

Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

​

?

291.25mm

19.49mm

Now we know one side of our spoke triangle and have the information we need to find another:

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The distance from the center of the hub to the end of the spoke is one half our ERD. Subtract the vertical edge we found to find the remaining length:

605/2-11.25=291.25mm

22.5mm

60°

11.25mm

302.5mm

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

​

Side View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

​

291.90mm

291.25mm

19.49mm

Now that we know two sides of the triangle we can use the Pythagorean theorem to find the hypotenuse:

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19.49² + 291.25² = 291.90²

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Unfortunately wheels are not two dimensional, so we are not done yet. Next we move on to the other dimension.

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

​

Front View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

​

From the side view we found the vertical component of this triangle.

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For the horizontal component of this triangle you want whatever offset you have from your hub flange to your spoke hole. In the case of offset or staggered drilled rims you will need to account for that. In this example we are assuming the rim is drilled on the centerline, so we are just using the measurement from the hub flange to centerline.

291.90mm

21mm

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

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Front View

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

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Now use the Pythagorean theorem one more time to find the distance from the center of the spoke hole on the hub to the end of the spoke:

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21² + 291.90² = 292.65²

291.90mm

21mm

292.65mm

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

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One last step:

We got the distance from the center of the hub hole to the end of the spoke, but spokes are measured from the edge of the J-bend. This J-bend sits up against the edge of the hole in the hub, so to get our final spoke length, subtract half the hub hole size from the last number we got:

292.65 - 1.3 = 291.35mm

There’s you’re final spoke length. Hopefully working through this exercise can help you understand how to approach less common spoke configurations that you might not be able to find a calculator for.

ERD: 605mm

Offset: 21mm

Ø (PCD) : 45mm

Spoke Hole: 2.6mm

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