Shafts are universal components of machines with rotating parts. In a standard automobile, each axle connecting the front and rear wheels is a shaft around which the wheel sets rotate while the car is in motion.

These types of shafts tend to be of uniform diameter, or thickness, meaning that each end of the shaft looks the same. But some shafts taper, or become thinner from one end the other, usually at a constant rate. The nature of the job typically determines the "steepness" of the taper, which can be expressed in units, degrees or both.

If you look at a tapered shaft from the side, it takes the form of a triangle, with a base and two identical sides coming toward a point. This makes the tapered shaft a rotating cone, and if the point is small, the force generated by the rotation is focused on a tiny area and can thus be very potent.

Most tapered shafts do not come to a point. Instead, they have a larger diameter (denoted ​D​ for calculation purposes) at one end and a smaller diameter (​d​) at the other. The distance between them is given as ​L​. Tapered shafts are expressed in terms of their ​taper ratio​, which is the change in diameter divided by the change in length, or

The boat propeller provides a primary example of a tapered shaft. These shafts have other material threaded along them, like screws, usually flayed out at the end to provide propulsive thrust against the resistance of the water. Most rotate clockwise; some boats have twin propellers that rotate in opposite directions.

Common levels of taper in propellers include 1:10 (that is, a one unit increase in diameter for every 10-unit increase in length), 1:12 and 1:16. Specialized power boats are often made to unusual specifications. TPF, or taper per foot, is the most common unit used in this industry.

The following example relies on a 1 in 8 taper ratio, which is not especially common.

Say you are given a propeller with a small diameter of 1.5 ft. If the length is 12 ft, what is the value of the larger diameter?

Here you have ​d​ = 1.5, ​L​ = 12, and a taper ratio of 1:8, better expressed as the decimal 0.125 (1 divided into 8). You seek the value of ​D​.

From the information above, the taper ratio, here 0.125, is equal to (​D​ − ​d​) / ​L​, so:

Multiplying each side by 12 gives

To find the angle in degrees of this taper (i.e. the 1 in 8 taper angle), simply take the inverse tangent (tan^-1 or arctan) of this angle, which is half of the ratio of the two diameters (since ​L​ divides the "triangle" of the propeller into two smaller identical right triangles) divided by L − the familiar "opposite over adjacent" defining tangent in basic trigonometry.

As you may notice, this is the same as the taper ratio. In this case, the inverse tangent is 1.5/12 = 0.125, and the associated angle, which you can determine by using a calculator or just a web browser, is 7.13 degrees.

If you need, say, an easy taper per foot to degrees converter or any sort of taper-per-foot calculator (or whatever units of measure your needs demand), you can find a bevy of these at your disposal online. See Resources for one such example.

If you're an advanced student who is clever with computer languages, you could even write a simple program that does the math.