The wild west. A small and dusty town. A bar. A derivative walks into the bar. All the functions scatter. In a dark corner, a lone function dares to stand his ground. The derivative walks up to the lone function and asks, “Who the hell do you think you are?” Without skipping a beat, the function, with eyes barely visible from under the brim of his hat, replies “I’m ex.” The derivative gives a small, knowing grin and says, “Today is not your lucky day, I’m d/dy.” – Original source unknown
So it’s been a while since whipping out the high school math books. I mean who needs that kind of grief in the real world?
Well, ladies and gents, for the first time ever, I’m going to show you a practical use for your trigonometry classes.
But fear not, I’ll be holding your hand through this, you just need to plug in the numbers.
Why This Is Useful
The best option for choosing a stem is to have yourself measured on your bike and get a stem that matches the angle and length of the one used for measurement. There is no guessing here, and it’s all worked out well.
But what if your bike is not quite right, or if you didn’t have it fitted, or after some time the “correct” fit just doesn’t quite cut it?
Amongst other things, the stem will need to be adjusted.
Simple?
No.
Doable?
Yes.
Because of the angles, every adjustment of the stem affects not only the height, but the distance from the top of the saddle to the handlebars.
How My Dilemma Helps You
I am 6’1″ (185mm) and my build gives me legs that are quite long, but not long arms. I’m like a Tyrannosaurus on wheels.
Ideally I’d be riding a custom built frame that’s just right for me, but I don’t have deep pockets for that kinda stuff, so standard consumer bikes is the only option. Fitting my body on a standard, production bike takes a bit of work, thought and adjustment.
My current mountain bike setup puts the handlebars about 79cm in front of the saddle (measured from where my sitting bones would rest). This is a position I have had on my bikes for over 10 years already, but after riding a small, folding bike that is only 72cm from seat to handlebar, and finding it quite comfortable, I need to make some adjustments to my bigger bike. The vertical difference between saddle and handlebar currently puts the handlebar 2cm below the saddle.
There is one other sure giveaway of being too stretched out. When riding for over three hours I often place the middle of my fingers on the handlebars, rather than the palm. This position eases strain in my back and just feels right.
Target: move handlebars 4cm closer to seat, and raise them by up to 2cm.
Ready?
Let’s go.
Preparation
First, some abbreviations:
- hl = horizontal length, the length of your stem measured directly forward, parallel to level ground (we’ll calculate this number)
- vh = vertical height, the height of your stem as measured directly upwards (we’ll calculate this number)
- sa = stem angle, the angle marked on the stem
- sl = stem length, the length marked on the stem
- hta = head tube angle, the angle of your head tube (we’ll measure this)
To measure the head tube angle:
- Take angle from manufacturers specifications (I did)
or
- Stand bike securely (lean against something, tie rubber bands around brake levers and handlebars to stop sliding)
- Find something long and straight (broom, etc)
- Drop line straight down the head tube to the floor
- Measure the head tube angle as shown in the diagram, between the floor and that line (you’ll need a protractor for this)
To measure the stem length:
- Check specifications (often written on the underside of the stem)
or
- Measure from the center of the head tube to the center of the handlebar.
The Math Part
You’ll need the online scientific calculator (or a real one) for these.
Calculate horizontal length and vertical height.
bignum = 90 – head tube angle + stem angle
horizontal length = cos(bignum) x stem length
vertical height = sin(bignum) x stem length
Original Stem Figures
My current figures are…
- stem angle = 25
- stem length = 120mm
- head tube angle = 72
so…
bignum = 90 – 72 + 25 = 43
horizontal length = cos(43) x 120 = 87mm
vertical height = sin(43) x 120 = 81mm
Find A New Stem
Now you just need to plug in the numbers of a new stem to find the one that is closest to your requirements.
To get 4cm extra height is quite a rise, so I’ll try out a 45 degree stem first. These are available in a range of lengths, this angle is probably right, but the length will make a big difference.
120mm stem with 45 degree rise:
bignum = 90 – 72 + 45 = 63
horizontal length = cos(63) x 120 = 54mm
vertical height = sin(63) x 120 = 106mm
So I’ve moved back 33mm, but moved up by 25mm.
I was hoping to move further back and don’t need that much more height.
Next…
100mm stem with 45 degree rise:
bignum = 90 – 72 + 45 = 63
horizontal length = cos(63) x 100 = 45mm
vertical height = sin(63) x 100 = 89mm
Overall, for my bike, they work out as shown in the table below.
| Stem | Horizontal Length | Vertical Height |
|---|---|---|
| 120mm, 25 degree | 87mm | 81mm |
| 120mm, 45 degree | 54mm | 106mm |
| 100mm, 45 degree | 45mm | 89mm |
Conclusion
Taking the time to get the right stem is worth it. However, this isn’t really needed any more and there are plenty of shops with adjustable stems in their stock. It’s much easier to use one of those to determine the right length, etc.
If you are in the market for a very expensive stem, I’d recommend just buying an adjustable one (not that expensive), and then spend a little less on the full stem when you’ve determined the right size and length.