The Geometry of Bicycle Geometry

Bike geometry is important for fit and handling. Fitting a bike is a process of measurement and trial-and-error, but once you have a bike that fits, you should be able to replicate that fit to a new bike.

Every aspect of a bike's geometry affects the fit. The frame has a Reach and Stack measurement, which are the horizontal and vertical distance between the bottom bracket center and the steerer tube center. These are important for figuring out if a bike can fit, but other variables come into play for dialing in a precise fit with stem and handlebar concerns.

The total reach on a bike is the horizontal distance between the saddle and the grip point. The total stack on a bike is the vertical distance between the saddle and the grip point. The frame stack/reach does not factor in the seat tube, stem, or handlebars, which obviously contribute greatly to the bicycle. In order to determine the contribution of these factors to the total reach and stack, we need to perform a bit of high school geometry.

The combination of stack, reach, and handlebar width will determine your back angle for a given bike, which is one of the more significant aspects of fit and handling.

Triangles

Fortunately, almost all of this devolves into basic triangle geometry.

All the angles in a triangle add up to 180 degrees. So if you know two angles (a) and (b), you can determine the third by subtraction:

180 = a + b + c

180 - a - b = c

If we know angles and some side lengths, then we can calculate the other side lengths and angles:

sin(angle) = opposite / hypotenuse

cos(angle) = adjacent / hyptoneuse

tan(angle) = opposite / adjacent

Steerer Tube

The first component that frame reach/stack miss out on is the effect of the steerer tube. The steerer tube projects up and back from the frame based on the head tube angle. As you add spacers to the steerer tube, this moves the stem's initial point up and back. 

This geometry is relatively easy to work out. We start with a right triangle, with a point A at the center of the steerer tube at the frame, and point B at the center of the steerer tube at the stem. If you draw a straight across from A and down from B, you'll have point C. The angle at point C is 90 degrees, and the angle at point A is the head tube angle.

The distance between AC is the additional stack given by the steerer tube, and the distance BC is the reach given by the steerer tube. We know two of the angles (it's a right triangle with C=90 degrees, and angle A is given by the Head Tube Angle), and the length of the hypotenuse, so we can calculate these factors using the following formula:

sin(head tube angle) = stack / measured distance

cos(head tube angle) = reach / measured distance

We can rearrange the formula to get the unknown variable alone.

measured distance * sin(head tube angle) = stack

measured distance * cos(head tube angle) = reach

Assuming a measured distance of 60mm and a head tube angle of 69 degrees, we get a 56mm stack and 21mm reach back.  Each 10mm spacer adds about 9mm of stack and 4mm of reach with a 69 degree head tube angle.. Note that these reach numbers are subtracted from the total reach, because the angle leans backwards.

Stem

With that out of the way, let's determine how much reach and stack are added by the stem. Stems typically have two properties: length and angle. So you might buy a 110mm stem with a 6 degree angle, and point it up. At first blush, you may want to just add 110mm to the frame reach and call it a day. However, that won't be accounting for the stem angle or the head tube angle.

We account for the head tube angle by adding 90-HTA to the stem angle. The angle that we are adding is also the third angle in the triangle where we figured out the stack/reach adjustment of the steerer tube. From there, it's the same right triangle math that gives us the stem measurement.

sin(adjusted stem angle) = stack / stem length

cos(adjusted stem angle) = reach / stem length

If we have a head tube angle of 70.5 degrees, a stem angle of 7 degrees, and a stem length of 90mm, then we have a reach contribution of 80mm and a stack contribution of 40mm. An additional 10mm of stem length adds about 9mm of reach and 4mm of stack.

Handlebars

This is where it starts to get tricky. The reach and stack added by a handlebar depends significantly on how the handlebar is setup. Road handlebars are pretty standard, but gravel and mountain handlebars have a lot of possible tweaks for individual comfort. As such, it's difficult to determine this from a spec sheet. You'll want to measure for yourself the reach that's added by the handlebars in your preferred configuration.

Since this is a measurement and not a calculation, there's not any math we need to do. Just add the reach/stack to the figures you've got from the frame, steerer tube, and stem.

Grip width is an important factor in determining back angle. Again, you'll want to actually measure this - the handlebar width spec is only accurate on flat bars with 0 rise/sweep/flare. Gravel bars, comfort mountain bars, and weird alt bars will all differ significantly.

Saddle

You can calculate this, but it's easier to just measure it, especially if you don't have a perfectly straight seat tube and seat post. You need to know the vertical distance between the bottom bracket and the point your sitbones sit, as well as the horizontal distance.

The difference in saddle height and frame/stem/handlebar stack is your total stack. The addition of horizontal saddle distance from bottom bracket and frame/stem/handlebar reach is your total reach.

Back Angle

This is the variable we want to control. Even for a specific person, you'll likely want a different back angle on your commuter/touring bike than on your road racing bike. Your snowy fat bike wants a more upright posture than your trail slaying squishy bike. So ultimately it's all personal preference!

This is where we start needing body measurements.

• Back length: sit bones to shoulder bones
• Shoulder width: shoulder bone to shoulder bone
• Arm length: shoulder bone to midpoint of palm (the grippy part)
• You probably don't want to ride with totally stretched out and extended arms. And maybe a more powerful fit analysis could measure your arm segment lengths and ideal elbow angle to determine the best handlebar setup. Meh. The back angle is subjective anyway so let's assume a totally straight arm.

This is also where we start getting into 3D geometry. But I'm going to cheat and simplify a bit so I only have to solve two dimensional problems.

Horizontal Shoulder Angle

Stand with your right arm straight out in front of you. This is a 'horizontal shoulder angle' of 0 degrees. The amount of reach in front of you that you have is simply your arm length. Now, rotate your arm to the right. As the horizontal shoulder angle grows, your reach directly in front of you decreases. Fortunately, it's easy to calculate this.

We'll start by drawing an isosceles trapezoid, which is a fancy set of words for "a box where the two sides are sloped at the same angle." The top line will be the shoulder width, and the bottom line is the handlebar width. These lines should be parallel. Then we connect the two ends together. If the handlebar width and shoulder width are identical, then we have a rectangle. Cool! You have an HSA of 0 degrees and can use your arm length unmodified in the next bits.

Otherwise, you're going to draw some triangles. Pick a point on the wider line that is directly above/below your narrower line's end, and draw a straight line to the narrower line. You now have a right triangle. You can throw away the rest of the drawing, because we're assuming that you have left/right symmetry, and we can now do calculations to figure out the angle.

The points in the triangle are:

    

    A. The narrow point (probably your shoulder endpoint)

    B. The wider point (probably your handlebar endpoint)

    C. The narrow point, projected down onto the wider line.

The angles are:

    A. Horizontal Shoulder Angle

    B. 

    C. 90 degrees.

The segment AB is known: your arm length. The segment BC is also known - we divide the shoulder width and handlebar width by two and add them together. This allows us to calculate the horizontal shoulder angle.

We apply the horizontal shoulder angle to the arm length to get the adjusted arm length.

All Variables Known

Now, we can describe a triangle of three segments:

1. Adjusted arm length (shoulder-grip)
2. Back length (butt-shoulder)
3. Saddle-to-grip length (butt-grip)

With these segments, we can determine the back angle relative to this triangle. When we have that angle, we can translate it with the angle between the saddle and the grip to determine the back angle relative to the ground.

Go ahead and plug all this into a spreadsheet, and start playing around with variables. It's fascinating to see how this all works out - specifically, that wider handlebars often call for either reduced reach or increased stack to retain the same back angle.