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The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along the slope) and rise is the vertical distance.

## Contents

Slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described as grades, but typically grades are used for human-made surfaces such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle routes. The grade may refer to the longitudinal slope or the perpendicular cross slope.

## Nomenclature

There are several ways to express 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 ${\displaystyle 100\times {\frac {\text{rise}}{\text{run}}}}$ which is equivalent to the tangent of the angle of inclination times 100. In Europe and the U.S. percentage "grade" is the most commonly used figure for describing slopes.
3. as a per mille figure (‰), the formula for which is ${\displaystyle 1000\times {\frac {\text{rise}}{\text{run}}}}$ 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. It is sometimes written as mm/m instead of the ‰ symbol. [1]
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 1000 feet of run would have a slope ratio of 1 in 200. (The word "in" is normally used rather than the mathematical ratio notation of "1:200".) This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s.
5. as a ratio of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, "slopes are expressed as ratios such as 4:1. This means that for every 4 units (feet or metres) of horizontal distance there is a 1 unit (foot or metre) vertical change either up or down." [2]

Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α by taking the inverse tangent of the standard mathematical slope, which is rise / run or the grade / 100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical.

Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse (the slope length). This is not the usual way to specify slope; this nonstandard expression follows the sine function rather than the tangent function, so it calls a 45 degree slope a 71 percent grade instead of a 100 percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that, in order to calculate the grade (100% × rise/run) or standard slope (rise/run). When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using the sine of the angle rather than the tangent) makes only an insignificant difference and can then be used as an approximation. Railway gradients are often expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In either case, the following identity holds for all inclinations up to 90 degrees: ${\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}}$. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage).

### Equations

Grades are related using the following equations with symbols from the figure at top.

#### Tangent as a ratio

${\displaystyle \tan {\alpha }={\frac {\Delta h}{d}}}$

The slope expressed as a percentage can similarly be determined from the tangent of the angle:

${\displaystyle \%\,{\text{slope}}=100\tan {\alpha }}$

#### Angle from a tangent gradient

${\displaystyle \alpha =\arctan {\frac {\Delta h}{d}}}$

If the tangent is expressed as a percentage, the angle can be determined as:

${\displaystyle \alpha =\arctan {\frac {\%\,{\text{slope}}}{100}}}$

If the angle is expressed as a ratio (1 in n) then:

${\displaystyle \alpha =\arctan {\frac {1}{n}}}$

### Example slopes comparing the notations

For degrees, percentage (%) and per-mille (‰) notations, larger numbers are steeper slopes. For ratios, larger numbers n of 1 in n are shallower, easier slopes.

The examples show round numbers in one or more of the notations and some documented and reasonably well known instances.

Examples of slopes in the various notations
DegreesPercentage (%)Per mille (‰)RatioRemarks
60°173%1732‰1 in 0.58
47.7°110%1100‰1 in 0.91 Stoosbahn (funicular railway)
45°100%1000‰1 in 1
30.1°58%580‰1 in 1.724 Lynton and Lynmouth Cliff Railway (funicular railway)
30°58%577‰1 in 1.73
25.5°47%476‰1 in 2.1 Pilatus Railway (steepest rack railway)
20.3°37%370‰1 in 2.70 Mount Washington Cog Railway (maximum grade)
20°36%363‰1 in 2.75
18.4°33%333‰1 in 3
8.13°14.2%142‰1 in 7
7.12°12.5%125‰1 in 8 Cable incline on the Cromford and High Peak Railway
4.0°7%70‰1 in 14.3
3.37°5.9%59‰1 in 17Swannington incline on the Leicester and Swannington Railway
2.86°5%50‰1 in 20 Matheran Hill Railway. The incline from the Crawlerway at the Kennedy Space Center to the launch pads. [4] [5]
2.29°4%40‰1 in 25 Cologne–Frankfurt high-speed rail line
2.0°3.5%35‰1 in 28.57 LGV Sud-Est, LGV Est, LGV Méditerranée
1.97°3.4%34‰1 in 29Bagworth incline on the Leicester and Swannington Railway
1.89°3.3%33‰1 in 30.3Rampe de Capvern on the Toulouse–Bayonne railway  [ fr ]
1.52°2.65%26.5‰1 in 37.7 Lickey Incline
1.43°2.5%25‰1 in 40 LGV Atlantique, LGV Nord. The Schiefe Ebene.
1.146°2%20‰1 in 50Railway near Jílové u Prahy. Devonshire Tunnel
0.819°1.43%14.3‰1 in 70 Waverley Route
0.716°1.25%12.5‰1 in 80Ruling grade of a secondary main line. Wellington Bank, Somerset
0.637°1.11%11.11‰1 in 90 Dove Holes Tunnel
0.573°1%10‰1 in 100 The long drag on the Settle & Carlisle line
0.458°0.8%8‰1 in 125Rampe de Guillerval
0.2865°0.5%5‰1 in 200 Paris–Bordeaux railway  [ fr ], except for the rampe de Guillerval
0.1719°0.3%3‰1 in 333
0.1146°0.2%2‰1 in 500
0.0868°0.1515%1.515‰1 in 660 Brunel's Billiard Table - Didcot to Swindon
0.0434°0.07575%0.7575‰1 in 1320 Brunel's Billiard Table - Paddington to Didcot
0%0‰1 in (infinity)Flat

In vehicular engineering, various land-based designs (automobiles, sport utility vehicles, trucks, trains, etc.) are rated for their ability to ascend terrain. Trains typically rate much lower than automobiles. 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 fills or cuts where these techniques have been used to create them.

In the United States, maximum grade for Federally funded highways is specified in a design table based on terrain and design speeds, [6] with up to 6% generally allowed in mountainous areas and hilly urban areas with exceptions for up to 7% grades on mountainous roads with speed limits below 60 mph (95 km/h).

The steepest roads in the world are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales [7] and Canton Avenue in Pittsburgh, Pennsylvania. [8] The Guinness World Record once again lists Baldwin Street as the steepest street in the world, with a 34.8% grade (1 in 2.87) after a successful appeal [9] against the ruling that handed the title, briefly, to Ffordd Pen Llech. The Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue. [10] The street has formed part of a bicycle race since 1983. [11]

The San Francisco Municipal Railway operates bus service among the city's hills. The steepest grade for bus operations is 23.1% by the 67-Bernal Heights on Alabama Street between Ripley and Esmeralda Streets. [12]

## Environmental design

Grade, pitch, and slope are important components in landscape design, garden design, landscape architecture, and architecture; for engineering and aesthetic design factors. Drainage, slope stability, circulation of people and vehicles, complying with building codes, and design integration are all aspects of slope considerations in environmental design.

## Railways

Ruling gradients limit the load that a locomotive can haul, including the weight of the locomotive itself. On a 1% gradient (1 in 100) a locomotive can pull half (or less) of the load that it can pull on level track. (A heavily loaded train rolling at 20 km/h on heavy rail may require ten times the pull on a 1% upgrade that it does on the level at that speed.)

Early railways in the United Kingdom were laid out with very gentle gradients, such as 0.07575% (1 in 1320) and 0.1515% (1 in 660) on the Great Western main line, nicknamed Brunel's Billiard Table, because the early locomotives (and their brakes) were feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage, such as the 1.2 kilometres (0.75 miles) section from Euston to Camden Town.

Extremely steep gradients require the use of cables (such as the Scenic Railway at Katoomba Scenic World, Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying funicular [13] ) or some kind of rack railway (such as the Pilatus railway in Switzerland, with a maximum grade of 48% (26°), claimed to be the world's steepest rack railway [14] ) to help the train ascend or descend.

Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in n, x% or y per mille. Since designers like round figures, the method of expression can affect the gradients selected.[ citation needed ]

The steepest railway lines that do not use a rack system include:

### Compensation for curvature

Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make the ruling grade uniform throughout, the gradient on those sharp curves should be reduced slightly.

### Continuous brakes

In the era before they were provided with continuous brakes, whether air brakes or vacuum brakes, steep gradients made it extremely difficult for trains to stop safely. In those days, for example, an inspector insisted that Rudgwick railway station in West Sussex be regraded. He would not allow it to open until the gradient through the platform was eased from 1 in 80 to 1 in 130.

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## References

1. Description of the Pau-Canfranc railway line - Operations (English Wikipedia) - Tracé (French Wikipedia)
2. Strom, Steven; Nathan, Kurt; Woland, Jake (2013). "Slopes expressed as ratios and degrees". Site Engineering for Landscape Architects (6th ed.). Wiley Publishing. p. 71. ISBN   978-1118090862.
3. "Traffic signs". www.gov.uk. The Highway Code - Guidance. Retrieved 26 March 2016.
4. "Crawler-Transporter". NASA. 21 April 2003. Archived from the original on 1 June 2020. Retrieved 18 June 2020.
5. "Countdown! NASA Launch Vehicles and Facilities" (PDF). NASA. October 1991. pp. 16–17. PMS 018-B, section 3. Archived from the original (PDF) on 27 January 2005. Retrieved 21 August 2013.
6. A Policy on Geometric Design of Highways and Streets (PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. 2001. pp. 507 (design speed), 510 (exhibit 8–1: Maximum grades for rural and urban freeways). ISBN   1-56051-156-7 . Retrieved 11 April 2014.
7. "Welsh town claims record title for world's steepest street". Guinness World Records. 16 July 2019.
8. "Baldwin street in New Zealand reinstated as the world's steepest street". Guinness World Records. 8 April 2020.
9. "Canton Avenue, Beechview, PA". Post Gazette.
10. "The steepest road on Earth takes no prisoners". Wired. Autopia. December 2010.
11. "General Information". San Francisco Metropolitan Transportation Agency. Archived from the original on 3 December 2016. Retrieved 20 September 2016.
12. "Top five funicular railways". Sydney Morning Herald.
13. "A wonderful railway". The Register . Adelaide, Australia. 2 March 1920. p. 5. Retrieved 13 February 2013 via National Library of Australia.
14. "The New Pöstlingberg Railway" (PDF). Linz Linien GmbH. 2009. Archived from the original (PDF) on 22 July 2011. Retrieved 6 January 2011.
15. "Pantele din Iaşi pun probleme ofertanţilor" (in Romanian). 5 March 2019.
16. "Return of the (modern) streetcar – Portland leads the way" (Press release). Tramways & Urban Transit. Light Rail Transit Association. October 2001. Archived from the original on 27 September 2013. Retrieved 15 December 2018.
17. "Boston's Light Rail Transit Prepares for the Next Hundred Years" (PDF). onlinepubs.trb.org. Retrieved 23 February 2021.
18. Daniel, Mac (11 November 2005). "Lechmere, Science Park stations reopen". Boston.com. Archived from the original on 6 March 2007. Retrieved 23 February 2021.
19. "Il Piano Tecnologico di RFI" (PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved 23 May 2019.