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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. 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: # as an ''angle'' of inclination to the horizontal. (This is the angle opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.) # as a ''percentage'', the formula for which is $100 \times \frac$ 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. # as a ''per mille'' figure (‰), the formula for which is $1000 \times \frac$ 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. # 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. # 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." Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle 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: $\tan = \frac$. 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). In Europe, road gradients are signed as a percentage.

Equations

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

Tangent as a ratio

:$\tan = \frac$ The slope expressed as a percentage can similarly be determined from the tangent of the angle: :$\%\,\text = 100 \tan$

:$\alpha = \arctan \quad$ If the tangent is expressed as a percentage, the angle can be determined as: :$\alpha = \arctan$ If the angle is expressed as a ratio ''(1 in n)'' then: :$\alpha = \arctan$

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

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.

* Aspect (geography) * Civil engineering * Construction surveying * Grading (engineering) * Cut-and-cover * Cut and fill * Cut (earthmoving) * Embankment (transportation) * Grade separation * Inclined plane * List of steepest gradients on adhesion railways. * Percentage * Per mille * Rake * Roof pitch * Ruling gradient * Slope * Slope stability ** Slope stability analysis * Surveying * Trench * Tunnel * Wheelchair ramp

References