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Bikexchange logo, link to Home    Bike MATH    Bikexchange logo, link to Home

By Patrick Clendenon

Editor note: Patrick, now a wise man of age 14, was 13 when he wrote this piece for school.


The first bike was made in France in 1790. It had no pedals. It was more like a wooden scooter. It had to be pushed and it could not be steered. In 1816 a German engineer came up with a steering mechanism. In 1860 a Scotsman improved the basic bike design by building a bike with both steering and pedaling systems. By 1885 an Englishman built the first chain-driven bicycle. In 1887 the first air-filled tire was introduced by an Irishman. Finally, as gearing systems were developed in the early part of this century, the basic design of the bike was complete.


The most important part of the bike is the frame, particularly it's geometry. The frame consists of a front and rear triangle which forms a parallelogram. The front triangle, also known as the main triangle, is actually a quadrilateral.

For the purpose of the design, the most important measurements are the head tube angle, the seat tube angle, the top tube length, and wheel base. These form the "design parallelogram." (See Frame Diagram .)

The head tube angle is formed by the intersection of the head tube line and the wheel base. The seat tube angle is formed by the intersection of the seat tube line and the wheel base. The wheel base is the distance from the center of the front wheel to the center of the rear wheel. In a true parallelogram, these angles are equal. In actual bike design, these angles are not exactly equal.

The angles depend on the type of bike and size of the rider. The racing bike has the least acute angles. The bigger the angle, the greater the speed. The road bike's angles are slightly more acute, so there is less speed, but more comfort. The mountain bike has the most acute seat and head tube angles. This allows for more comfort over rough, bumpy surfaces. Mountain bikes are not designed for speed.


Wheels come in all sizes, depending on the size of the frame. The larger the size of your wheel, the more distance you get per rotation of the wheel. We can prove this by using the formula for the circumference of a circle (Pi x Diameter). If 26 inches is the diameter of a wheel on a bicycle, using the formula for the circumference of a circle, we find out that (26 x 3.14 = 81.64) 81.64 inches is the distance around the wheel. So, when the rider has rotated his wheel once (one revolution), we know the rider has moved 81.64/12 = 6.8 feet.

With a bike computer, you have to enter the wheel diameter. The computer then uses Pi x D to calculate revolutions per minute (RPMs) and miles per hour (MPH).


The gear ratio is the number of teeth on the front sprocket divided by the number of teeth on the rear sprocket. When riding is easy the gear ratio is lower. When riding is hard, the gear ratio is higher. In other words, you change the gear ratio based on the difficulty of the ride. On a hill, for example, if the front sprocket gear of a mountain bike has 60 teeth and rear sprocket gear has 20 teeth, then the gear ratio would be 60:20 or 3:1. This means that the front sprocket revolves once for every time the rear sprocket revolves three times. On the other hand, if you are riding on a flat road, you could be using gears with the same number of teeth in both sprockets. For example, 30:30 would be a 1:1 gear ratio, common when riding fast.

As you can see, there's plenty of Math in biking!

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