Slope Gradient Calculator
Free Slope gradient Calculator for geomorphology & mapping. Enter variables to compute results with formulas and detailed steps.
Formula
Slope = Rise/Run; Angle = atan(Rise/Run); Percent = (Rise/Run)*100
Rise is vertical change, Run is horizontal distance, slope distance = sqrt(Rise^2 + Run^2).
Worked Examples
Example 1: Road Engineering Gradient
Problem: A road rises 150 m over 800 m horizontal. Calculate all slope expressions.
Solution: Slope ratio = 150/800 = 0.1875\nAngle = atan(0.1875) = 10.62 deg\nPercentage = 18.75%\nSlope distance = sqrt(800^2+150^2) = 813.94 m\nRatio = 1:5.33
Result: Slope: 18.75% | Angle: 10.62 deg | Ratio 1:5.33 | Dist: 813.94 m
Example 2: Hillslope from Contour Map
Problem: Point A at 520 m and B at 670 m, separated by 800 m horizontal.
Solution: Elevation diff = 150 m\nSlope = 150/800 = 0.1875\nAngle = 10.62 deg\nGrade = 18.75%\nSlope dist = 813.94 m
Result: Slope: 18.75% | Angle: 10.62 deg | Slope Distance: 813.94 m
Frequently Asked Questions
What is slope gradient and how is it expressed?
Slope gradient measures terrain steepness as the rate of elevation change with horizontal distance. It is expressed as a ratio of rise to run, as degrees from horizontal, as a percentage where 100% equals 45 degrees, as per mille for gentle gradients like railways, or as 1:n meaning one unit vertical for n units horizontal. Engineers typically use percentage for roads, surveyors prefer degrees, and geomorphologists use all forms. Understanding relationships between expressions is essential for communicating terrain information across disciplines.
How do you calculate slope from two elevation points?
Slope between two points is calculated by dividing the elevation difference by horizontal distance: S = delta-h / d. The horizontal distance must be the map distance, not slope distance along the ground. To convert to degrees, take the arctangent. To express as percentage, multiply by 100. When working with topographic maps, horizontal distance is measured using the map scale while elevation difference is from contour lines. In GIS, slope is computed automatically from a DEM using elevation differences between neighboring cells.
What is the difference between slope angle and percentage?
Slope angle is measured in degrees from horizontal, ranging from 0 to 90, while slope percentage is rise divided by run times 100. The key difference is percentage has no upper limit: 45 degrees equals 100%, 60 degrees equals 173%, and a vertical cliff approaches infinity. This nonlinear relationship means at gentle slopes the two are nearly proportional but diverge dramatically on steep terrain. Percentage is intuitive for road design where 6% grade means 6 m rise per 100 m horizontal. Angle is preferred in geomorphology where steep terrain is common.
How is slope calculated from a DEM?
In a DEM, slope at each cell is calculated from surrounding cell elevations using finite difference algorithms. The most common is Horn 1981, using all eight neighbors in a 3x3 window to compute partial derivatives in east-west and north-south directions. Slope magnitude is then sqrt(dz/dx squared + dz/dy squared). This algorithm is more robust to noise than simpler two-point methods. Computed slope depends on DEM resolution: coarser resolution smooths terrain and produces lower maximum slope values.
What are typical slope values for different terrain types?
Alluvial plains and coastal lowlands have slopes below 2% or about 1 degree. Rolling agricultural land ranges 2 to 8%. Moderate hillslopes for grazing range 8 to 20% or 5 to 11 degrees. Steep forested hillslopes range 20 to 45% or 11 to 24 degrees. Mountain slopes exceed 45% and cliffs approach vertical. Most soils become unstable above 35 to 40 degrees, the angle of repose for most unconsolidated materials. Understanding these ranges helps in land capability assessment and hazard zoning.
How do you convert between slope expressions?
To convert ratio to degrees: arctan(rise/run). Degrees to percentage: tan(angle) * 100. Percentage to degrees: arctan(pct/100). The ratio 1:n means percentage = 100/n and angle = arctan(1/n). Per mille is ratio times 1000. For gentle slopes below 5 degrees, angle and percentage are approximately equal. A useful reference: 1 degree approximately equals 1.75 percent gradient. These conversions are essential for engineers and geomorphologists working across different conventions.