About

In civil engineering, specifically railway design and road grading, slopes are frequently expressed in permille (‰) rather than percentages or degrees. This notation represents the change in elevation per 1000 units of horizontal distance. Converting this to degrees is essential for surveying equipment setup and verifying compatibility with vehicle climbing tolerances.

This tool visualizes the gradient using a dynamic vector graphic, helping engineers verify the physical steepness. It utilizes the inverse tangent function, arctan, to map the ratio h/1000 to an angle. The inverse function is included for surveyors converting raw angular data back into project documentation standards.

Formulas

Permille (‰) is defined as rise over 1000 run.

Deg = arctan(Permille1000)

To convert Degrees back to Permille:

Permille = tan(Deg) × 1000

Reference Data

Slope (‰)	Slope (°)	Common Application
0 ‰	0.00°	Level track
2.5 ‰	0.14°	Standard drainage minimum
10 ‰	0.57°	Mainline railway max (1%)
12.5 ‰	0.72°	High speed rail limit
25 ‰	1.43°	Steep branch lines
40 ‰	2.29°	Adhesion railway limit (approx)
100 ‰	5.71°	Rack railway territory
1000 ‰	45.0°	1:1 Slope

Frequently Asked Questions

Percent (%) is parts per 100. Permille (‰) is parts per 1000. A slope of 1% is equal to 10‰. Railways prefer permille because grades are often less than 1% and decimals can be cumbersome.

For very small slopes (under 20‰), the slope in radians roughly equals the rise/run ratio. However, this tool calculates the exact geometry using arctan to ensure precision at steeper grades.

Rarely. Roofing typically uses Pitch (rise/12) or Degrees. Permille is almost exclusively used in civil works (roads, tunnels, rails) and fluid dynamics gradients.

Physically, a vertical wall is infinite permille (90°). Inputs above 10000 are generally treated as near-vertical, but calculation remains valid.