A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or "rise over run". It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers correlate to more vertical inclinations. There are three common numbering systems: Look up Slope in Wiktionary, the free dictionary. ... A percentage is a way of expressing a proportion, a ratio or a fraction as a whole number, by using 100 as the denominator. ... For other uses, see tangent (disambiguation). ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... The panoramic view from Connors Hill, near Swifts Creek, Victoria A hill is a landform that extends above the surrounding terrain, in a limited area. ... Butchers Creek, Omeo, Victoria A stream, brook, beck, burn or creek, is a body of water with a detectable current, confined within a bed and banks. ... The roofs of Olomouc, Czech Republic. ... Rail tracks. ... Mountain road with hairpin turns in the French Alps For other uses, see Road (disambiguation). ... “Gravity” redirects here. ...

• the angle from horizontal in degrees,
• as a percentage: the tangent of the angle of inclination: the ratio of the altitude change to the horizontal distance (this is the more common percentage type), or
• an alternative definition as a percentage: the sine of the angle: the ratio of the altitude change to the surface length between any two points on the grade—also known as rise to run (not to be confused with the "rise over run" taught in grade-school geometry).

The difference between the latter two is small for gentle slopes (see small-angle formula). The ambiguities and the small differences that result may permit these two inconsistent approaches to coexist unrecognized, especially where grades considered are 15% or less. For other uses, see tangent (disambiguation). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... The small-angle formula is a mathematical approximation, used in astronomy. ...

Many of the mathematical principles of slope, that follow from the definition, are applicable in topographic practice. Grade is usually expressed as a percentage. Expressing it as the angle from horizontal carries the same information, but may lead to confusion for readers who are not proficient in trigonometry: they may confuse degree with percent, and/or not know how to do the conversion. In the UK, for road signs, maps and construction work, the gradient is often expressed as a ratio such as 1 in 12, or as a percentage [1]. Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Unused traffic signs in Austria Most countries post signage, known as traffic signs or road signs, at the side of roads to impart information to road users. ... For other uses, see Map (disambiguation). ... This article does not cite any references or sources. ...

In vehicular engineering, various land-based designs (cars, SUVs, trucks, trains, etc.) are rated for their ability to ascend terrain. (Trains typically rate much lower than cars.) The highest grade a vehicle can ascend while maintaining a particular speed is sometimes termed that vehicle's "gradeability" (or, less often, "grade ability"). The lateral slopes of a highway geometry are sometimes called fill or cuts. The Trikke is a Human Powered Vehicle (HPV) Automobiles are among the most commonly used engine powered vehicles. ... Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... A landform comprises a geomorphological unit, and is largely defined by its surface form and location in the landscape, as part of the terrain, and as such, is typically an element of topography. ... “Car” and “Cars” redirect here. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... For other uses, see Truck (disambiguation). ... For other uses, see Train (disambiguation). ... Look up ability in Wiktionary, the free dictionary Ability - the quality of person of being able to perform; A quality that permits or facilitates achievement or accomplishment. ... In order to keep a road or rail line straight and/or flat, and where the comparative cost or practicality of alternate solutions (such as diversion) is too prohibitive, the land over which the road or rail line will travel is built up to form an embankment. ... In order to keep a road or rail line straight and/or flat, and where the comparative cost or practicality of alternate solutions (such as diversion) is too prohibitive, a piece of a hill or mountain is cut out to make way for it. ...

Railways

Steep gradients limit the amount of load that a locomotive can haul, including the weight of the locomotive itself. A 1 in 100 gradient (1%) halves the load. Early railways in the United Kingdom were laid out with very gentle gradients, such as 1 in 2000 because the early locomotives (and their brakes) were so feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage, such as from Euston to Camden Town, about 8 km. Extremely steep gradients need the help of cables, or some kind of rack railway. Facade of Euston Station, London Euston Arch: the original Euston Station, as enlarged, ca 1851 Euston station concourse Euston station (also known as London Euston), is a large railway station in Central London. ... Camden Town is an area of North London, England, in the London Borough of Camden. ... Rack railway track using Von Roll system rack. ...

The steepest non-rack railway lines include:

• 1 in 18 - Flåm, Norway.
• 1 in 25 - Tarana - Oberon, New South Wales.
• 1 in 33 - Valley Heights - Katoomba, New South Wales
• 1 in 40 - Goulburn - Cootamundra, New South Wales (southbound)

Flåm is a small town situated at the end of Sognefjorden. ... Tarana is a small town in the mid-west of New South Wales, Australia. ... Oberon, also Auberon, King of Shadows and Fairies, is best known as a character in William Shakespeares play, A Midsummer Nights Dream, written in the mid-1590s. ... “NSW” redirects here. ... Valley Heights is a suburb of the City of the Blue Mountains. ... Katoomba is the chief town of the City of the Blue Mountains in New South Wales, Australia and is its administrative headquarters. ... “NSW” redirects here. ... Goulburn (New South Wales, Australia) is a quiet provincial city characterised by a particularly long main street. ... Cootamundra is a town and Local Government Area in the Riverina region of New South Wales, Australia. ... “NSW” redirects here. ...

See also

• Grade separation
• Slope

An example of a four-level stack interchange in the Netherlands. ... Look up Slope in Wiktionary, the free dictionary. ...

External links

• A Simple Explanation and Example of Grade Calculation