# Spoke Length Calculation

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## Welcome

Roger Musson’s “Professional Guide to Wheel Building” is a must read for anyone interested in learning how to build their own wheels. 15 USD is the cost of entry and a bargain for the content alone. Roger encourages staying up to date on the art of wheel building by offering free downloads of updated editions. Follow the first link to purchase the 6th edition of Roger’s masterpiece. I owe Sheldon Brown a huge thank you. The shear volume of work he contributed to the cycling community is astounding. Every article he published breathes Sheldon’s love for the bicycle. Sheldon is one of my hero’s and a shining example of what is means to live as an individual. I would not have written this if it were not for his selfless contribution to the bicycling community. Sheldon lost his life from a heart attack on February 4, 2008. He had been diagnosed with primary progressive multiple sclerosis in August of 2007.

http://sheldonbrown.com/wheelbuild.html

For some, necessity is the mother of all invention. For me, necessity is the mother of all learning. I was sitting on the side of the road, stranded because I had broken a spoke. This is where the learning would begin. It had been a rough winter and I was looking forward to getting back into top form. Nothing was going to stop me, except this broken spoke. This particular wheel had problems because this was just the first of several broken spokes that left me stranded on the side of the road. At some point one too many spokes broke and so did my patience. Which is to say I cut in two the remaining spokes out of frustration. This meant I had no idea which ones were drive side, which ones were leading and which one were trailing. I now have a spokeless wheel and a world of exploration at my fingertips to try and figure out how the hell to rebuild what I destroyed.

I didn’t have the money or the desire to buy a new wheel. So I went about scouring the internet looking for a solution. This is how I met Sheldon Brown and Harris Cyclery. Google “how to build a bicycle wheel” and Sheldon is #1 on the list of results. I found out how to build a wheel and oh so much more. If you like bicycles I encourage you to visit Sheldon’s site. If you LOVE bicycles like I do, your life will never be the same after visiting http://www.sheldonbrown.com/. I also was learning about the online treasure trove known as “eBay”. I fancied myself as a mathematician, which is hysterical because my math is only slightly better than my Spanish and I don’t speak Spanish.

Ok, maybe my math isn’t that bad, it was at least good enough to get me a job teaching Algebra and Trigonometry at Pamlico Community College. Most of my students had not gone to college, most were older than me and almost all of them were out of work. They often referred to me as, *“that crazy fool who rides his bicycle 26 miles to work.”* How is this crazy fool going to teach them Algebra? I was having visions of the great Eric Cartmenez when he said and I quote,*“How do I reeeach deeeze keeeeeedz?” *I decided this was to be the jumping off point for my students taking Intro to College Mathematics. When I came into class here is the equation I wrote on the board.

$ Spoke Length = \sqrt {R^2 + H^2 + F^2 – 2{RH} \cos \frac {720}{h}\cdot X } – \frac {\phi}{2} \qquad $

Where:

$ R = Rim radius to spoke ends based on ERD (effective rim diameter) $

$ H = Hub flange radius to spoke holes $

$ F = Flange offset $

$ h = Holes in rim $

$ X = Cross pattern $

$ \phi = Diameter of Spoke Hole in Hub $

I started writing with my left hand and mid way I switched to my right hand. The illusion of ambidexterity solidified my mastery of mathematical methods. This formula looks complicated and sophisticated but it is simple to solve and great for teaching the order of operations, a fundamental principle of algebra. PEMDAS, “pim-das”, or as I prefer, Please Excuse My Dear Aunt Steven, she likes to think of herself as a man. May you never forget. If you have forgotten then go here for a refresher. Now, back to the equation.

$ Spoke Length = \sqrt {R^2 + H^2 + F^2 – 2{RH} \cos \frac {720}{h}\cdot X } – \frac {\phi}{2} \qquad $

Oh yeah, that’s a nasty looking beast. Especially this part $ \big(\cos \frac {720}{h}\cdot X)\qquad $ and then when I tell you to we will need to divide $ cos \big(\frac{720}{h} \cdot X \big) $ by $ \big(\frac{360}{2\pi}\big) $ giving us a monstrosity that looks something like this $ cos\bigg (\frac{\frac{720}{h} \cdot X}{\frac{360}{2\pi}}\bigg) $ Your hands will tremble, your brow will sweat and you will give up all hope of becoming anything other than a basement dwelling urchin. Fear not! All we need to do is break it down into manageable chunks. This is a powerful strategy be prepared for an infusion of bone crushing confidence.

Here are the numbers I will use for this demonstration.

$ R = 270$

$ H = 19$

$ F = 36$

$ X = 3$

$ h = 32$

$ \phi = 2.3 $

First, plug all the numbers into the variables. You should have something that looks like this. The periods suspended in mid air $ \cdot$ serve the same purpose as the more familiar $ \times $ symbol for multiplication.

$ \sqrt {270^2 + 19^2 + 36^2 – 2\cdot 270\cdot19 \cos \big(\frac{720}{32}\times 3} \big) – \frac {2.3}{2} \qquad $

Notice where we stop “square rooting” this formula. The minus sign at the end is the first operator outside the square root sign. This means everything to the left will fall victim to the “squaring of the root” and everything to the right shall be spared, for now.

So lets dive back under the roof of the square root sign and tackle the first set of parantheses $cos \big({\frac{720}{h} \cdot X}\big)$. This expression says to $\div 720$ by $32$ and then $\times$ the result by $3$ which gives us $67.5$. Then we will take the $\cos$ of that *EXCEPT* we want this result in radians and not degrees. Why? Because Microsoft Excel uses radian to make trigonometric calculations, not degrees. In the next post I will show you how to set up this formula in Excel. To convert to radians you can either $\times$ the result by $\big( \frac {2\pi}{360} \big)$ or $\div$ by the inverse which would be $\big( \frac {360}{2\pi} \big)$. I chose the latter so multiply $2 \times \pi$ and $\div$ by $360$ which returns $57.325$. *(Note: I am lazy and used $3.14$ instead of $\pi$, if you used $\pi$ you may get a result that is slightly different and also inconsequential to the task at hand.)* Now all we do is $\div 67.5$ by $57.325$ to get our result in radians. That result is $1.1775$ and the $\cos$ is $0.3823515$. Looking more manageable!

$ = \sqrt {270^2 + 19^2 + 36^2 – 2\cdot 270\cdot19 \times 0.38 } – \frac {2.3}{2} \qquad $

Lets finish up what remains under the roof.

$ \Big(\big(270^2 + 19^2 + 36^2\big) – \big(2\cdot 270\cdot19 \times 0.38\big)\Big)$

$ \big(270^2 + 19^2 + 36^2\big) = 74,557 $

$ \big(2 \times 270\times 19 \times 0.38\big) = 3,932 $

$ 74,557-3,932=70,625 $

Our menacing beast has been reduced (pun intended) to this:

$ = \sqrt 70,625 – \frac{2.3}{2} $

Lets go ahead and get rid of the $ \surd $ leaving us with $ 265.75 – \frac{2.3}{2} $ Now all that is left is to $ \div 2.3 $ by $ 2 $ which returns $ 1.15 $. The solution to our formula.

$ 265.75 – 1.15 = 264.60 $

You are now armed with the knowledge of spoke length. Head over to your local bike shop and ask for spokes that are 264.60 mm in length. Next time I will show you how to write this in an Excel friendly format. Using Excel is a faster and more accurate way to calculate spoke length. Remember, wheels that use disc brakes and all rear wheels are asymmetrically dished. Only front wheels without disc brakes are symmetrically dished. This means the spokes on the left are the same length as the ones on the right. A mountain bike with disc brakes on the front and rear will have at least 4 different spoke lengths to accommodate the offset flanges required to mount the disc rotors to the hubs and also to accommodate the cassette on the rear that carries the sprockets, which means solving this formula 4 times! Excel will make this task easy if not enjoyable.