Grade (slope) - Expression Nomenclature

Expression Nomenclature

There are several systems for expressing slope:

1. as an angle of inclination to the horizontal. (This is the angle α opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)
2. as a percentage, the formula for which is which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., this percentage "grade" is the most commonly used unit for communicating slopes in transportation (streets, roads, highways and rail tracks), surveying, construction, and civil engineering.
3. as a per mille figure, the formula for which is which could also be expressed as the tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway.
4. as a ratio of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20. (The word "in" is normally used rather than the mathematical ratio notation of "1:20"). This is generally the method used to describe railway grades in Australia and the UK.

Any one of these expressions may be used interchangeably to express the characteristics of a slope. Grade is usually expressed as a percentage, but this may easily be converted to the angle α from horizontal since that carries the same information.

There is a method in which slope may be expressed when the horizontal run is not known: rise divided by the hypotenuse (the slope length). This is not a usual way to measure slope. This follows the sine function rather than the tangent function and this method diverges from the "rise over run" method as angles start getting larger (see small-angle formula).

Many of the mathematical principles of slope that follow from the definition are applicable in topographic practice. In the UK, for road signs, maps and construction work, the gradient was traditionally expressed as a ratio such as 1 in 12, but signs showing gradient expressed as a percentage are becoming more common.

In civil engineering applications and physical geography, the slope is calculated along a particular direction of interest which is normally the route of a highway or railway road bed.