Shoulda paid more attention in school
It can be rare that anything you learned in school is ever required in your daily life, especially with regards to cycling. But when it comes to bike fit and figuring out which stem you need to get you a certain reach to the bar or drop from your saddle to bars, a little geometry can go a long way. When it comes to figuring out the rise/reach for different stems on different frames you might be wishing you didn't fall asleep so much in geometry class.
Things would be easy if everything were at 90degree angles, but they are not. You've got the head angle of your frame. Usually 73 for road bikes and 71 for mountain bikes. Then you have the angle of the stem. Note that the stem goes on top of the fork steerer tube that is sticking in your head tube that is at the angle just mentioned. So if you wanted your stem to be horizontal like the cool looking racers then you actually need a stem with a -17 rise (assuming a 73 head tube of a road bike)
Most mountain bike stems have 5 or 6 degree rises but there are 15-20 degree rises as well.
You need to know two things, the reach and the rise. Reach is the horizontal length of the stem, it is not the actual length of the stem, because you have to factor in the stem angle and the head tube angle. It's all a bunch of triangles.
Here is a sweet little chart/picture that helps a ton, courtesy of
If you want to calculate it on your own you can do this
Reach = COS (90degrees + stem rise - head angle)
One thing I'll always remember from Mr. Williams math class.
SOH - CAH - TOA
To be said like an American Indian Warrior with spear held high.
Sine= Opposite over Hypotenuse
Cosine = Adjacent over Hypotenuse
Tangent = Opposite over Adjacent