# Slope (definition, formula, examples)

In the exact sciences, such a characteristic is often used as a slope, which is also known by the names "slope", "scree" and "rise". The slope characterizes the lines and faces of flat and three-dimensional figures, for example, the hypotenuse in triangles or the sides in prisms.

In everyday life, slope is most often used to calculate flat surfaces and structures: roads, ramps, pipelines, stairs, and so on. The main units of measurement are degrees, percentages and ppm. The larger the numerical value, the stronger the slope, and vice versa.

From a mathematical point of view, the slope is the ratio of the vertical elevation to the horizontal position, or the ratio of the projection of the line in question in the vertical plane to the projection of this line on the horizontal surface.

In addition to the slope itself, the lengths of surfaces can be taken into account: in millimeters, centimeters, meters, and so on. For example, if a road sign says "steep 10%", it means that the road will rise from the horizontal plane by 10 meters every 100 meters, or 100 meters every kilometer.

## Methods of Expressing Value

Slope can be expressed in different units: degrees, percentages, ppm, and proportions. Consider all options:

**In degrees.**The value is treated as a tilt/elevation angle to the horizontal. If we imagine the latter as the base of a right triangle, the slope will be equal to its hypotenuse. Accordingly, the angle can be calculated using the cosine formula: as the ratio of the adjacent leg to the hypotenuse, and expressed in degrees and fractions of a degree.**Percentage**A measurement system most commonly used in Western countries: USA and Europe. The slope in this case is calculated as a tangent (the ratio of the opposite leg to the adjacent one) multiplied by 100. The final value is always less than a hundred, and is expressed as a percentage.**In ppm.**This is a measurement system similar to the previous one, but with the tangent multiplied by 1000 instead of 100. It is used to describe / measure small slopes: railways, bridges, ramps, freeways, etc. Further. The unit of measure is ‰, or alternatively mm/m.**In proportion: as the ratio of one part of the climb to several parts of the run.**For example, if the road rises 5 meters every 1000 meters, its slope is indicated as 5:1000, or after shortening - 1:200 . This system of measures is widely used in the UK, Australia and Hong Kong.**In proportion: as the ratio of parts of run to one part of ascent.**This measurement method is the reverse of the previous one, and if the road rises 3 meters every 1000 meters, the slope will be indicated as 1000:3. The distance/length unit does not matter, and it can be either a meter (centimeter, millimeter) or a foot, inch, mile.

In this case, degrees and percent/ppm are values that need to be converted to each other. For example, a tilt angle of 45 degrees can be represented as 71%, and vice versa. When calculating, the tangent and sine formulas are used, and for small values of the angle, the length of the slope can be neglected, expressing it as an approximation. If the difference between the sin and tg of the angle is significant, the drawn tangent is used.

## Notation examples

Different countries use different ways to calculate and designate slopes. Percentages and ppm are more commonly used for gentle surfaces, and degrees and proportions for steep slopes. The following values can be given as examples of the designation of slopes:

- Stoosbahn funicular railway - 60 degrees or 173% or 1:0.58.
- Pilatusbahn railway - 25.5 degrees or 47% or 1:2.1.
- Waverley Route - 0.819 degrees or 1.43% or 1:70.
- Derbyshire Dove Holes Tunnel - 0.637 degrees or 1.11% or 1:90.
- Mount Washington Cog Railway - 14 degrees or 25% or 1:4.

The second indicator (percentage) can always be represented as ppm. So, 25% = 250‰, 47% = 470‰. The prevailing slope reduces the maximum load that a wheeled vehicle can carry. For example, a diesel locomotive needs 2 times more power to transport a train on rails with an inclination of 1% (or 1:100) than on a perfectly flat plane. Therefore, the slope calculation is mandatory for all roads and railways.

In the case of degrees, the slope is calculated using the tangent (dividing the height by the length), and to express it as a percentage and in ppm, you need to multiply h/L by 100 and 1000, respectively. The units of L and h must be the same, the value will be incorrect if the first, for example, is expressed in meters, and the second in feet.