Grade (slope)

The grade or gradient of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal or its tangent. 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 in which run is the horizontal distance and rise is the vertical distance.
Slopes of existing physical features such as canyons and hillsides, stream and river banks, and beds are often described as grades, but typically the word "grade" is 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:

as an angle of inclination to the horizontal.
as a percentage, the formula for which is, which is equivalent to the tangent of the angle of inclination times 100. At a slope angle of 45 degrees, the run is equal to the rise. Expressed as a percentage, the slope of this angle is 100 percent. As the slope approaches 90 degrees, the percentage slope approaches infinity. In Europe and the U.S. percentage grade is the most commonly used figure for describing slopes.
as a ‰ 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. It is sometimes written using mm/m or m/km instead of the ‰ symbol.
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. 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.
as a ratio of many parts run to one part rise, which is the inverse of the previous expression. For example, a slope expressed as 4:1 in this system means for a given vertical distance the horizontal distance travelled is four times as long.

Any of these may be used. When the term grade is used, the slope is usually expressed as a percentage. 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 at vertical.
Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse. 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 or standard slope. When the angle of inclination is small, using the slope length rather than the horizontal displacement 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:
. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the slope or the grade.
In Europe, road gradients are expressed in signage as percentage.

Equations

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

Tangent as a ratio

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

Angle from a tangent gradient

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

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.

Degrees	Percentage	Permillage	Ratio	Remarks
60°	173%	1732‰	1 in 0.58
47.7°	110%	1100‰	1 in 0.91	Stoosbahn
45°	100%	1000‰	1 in 1
35°	70%	700‰	1 in 1.428
30.1°	58%	580‰	1 in 1.724	Lynton and Lynmouth Cliff Railway
30°	58%	577‰	1 in 1.73
25.5°	47%	476‰	1 in 2.1	Pilatus Railway
20.3°	37%	370‰	1 in 2.70	Mount Washington Cog Railway
20°	36%	363‰	1 in 2.75
18.4°	33%	333‰	1 in 3
16.9°	30%	300‰	1 in 3.3	Extremely steep road
14.0°	25%	250‰	1 in 4	Very steep road. Mount Washington Cog Railway
11.3°	20%	200‰	1 in 5	Steep road
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
5.71°	10%	100‰	1 in 10	Steep road
4.0°	7%	70‰	1 in 14.3
3.37°	5.9%	59‰	1 in 17	Swannington 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.
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 29	Bagworth incline on the Leicester and Swannington Railway
1.89°	3.3%	33‰	1 in 30.3	Rampe de Capvern on the
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 50	Railway 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 80	Ruling 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 125	Rampe de Guillerval
0.2865°	0.5%	5‰	1 in 200	, 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 TableDidcot to Swindon
0.0434°	0.07575%	0.7575‰	1 in 1320	Brunel's Billiard TablePaddington to Didcot
0°	0%	0‰	1 in ∞	Flat

Roads

In vehicular engineering, various land-based designs 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". 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, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds, 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.
The steepest roads in the world according to the Guinness Book of World Records are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales and Canton Avenue in Pittsburgh, Pennsylvania. The Guinness World Record once again lists Baldwin Street as the steepest street in the world, with a 34.8% grade after a successful appeal against the ruling that handed the title, briefly, to Ffordd Pen Llech.
A number of streets elsewhere have steeper grades than those listed in the Guinness Book. Drawing on the U.S. National Elevation Dataset, 7x7 identified ten blocks of public streets in San Francisco open to vehicular traffic in the city with grades over 30 percent. The steepest, at 41 percent, is the block of Bradford Street above Tompkins Avenue in the Bernal Heights neighborhood. 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.
Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% for Canton Avenue. The street has formed part of a bicycle race since 1983.