Grade (slope)
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 gradient in calculus where zero indicates gravitational level. 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 and rise is the vertical distance.
The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described. Grades are typically specified for new linear constructions (such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle circulation routes). The grade may refer to the longitudinal slope or the perpendicular cross slope.
There are several ways to express slope:
Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α from horizontal or the other expressions.
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; it 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. 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. Railway gradients are usually 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 any case, the following identity holds for all inclinations up to 90 degrees:
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