Slope Inclination Calculator
Gradient · Slope · Angle · Elevation · Road Engineering · Drainage
Slope Calculator
Formulas & References
i = Δh / L [-]
Δh = Elevation difference; L = Horizontal distance; i = dimensionless ratio
i [%] = (Δh / L) · 100 [%]
Example: i = 0.05 → 5% grade or 1:20
α = arctan(i) [rad] or [°]
α ≈ i for small angles (i < 0.1)
Δh = i · L [m]
Calculation for known gradient and distance
L = Δh / i [m]
Calculation for known gradient and elevation
Typical Grade Values (Road Engineering / Drainage)
| 0.3% (1:333) | Flat road, minimal grade |
| 1–2% | Normal road, drainage, typical longitudinal grade |
| 2–4% | Road in slope, highway exit |
| 4–6% | Mountain road, steep drainage |
| 8–10% | Very steep road, mountain path |
| >10% | Extreme slope, specialized technique required |
• Elevation difference Δh = 5 m
• Horizontal distance L = 100 m
• Gradient i = 5 / 100 = 0.05 = 5% = 1:20
• Angle α = arctan(0.05) ≈ 2.86°
Technical Background
Slope and Inclination – Fundamentals of Surveying and Road Engineering
Slope (inclination) describes the steepness or grade of terrain, a road, or a drainage system. It is expressed as a dimensionless ratio and is fundamental to engineering projects:
- Road Engineering: Longitudinal and transverse grades for safe driving and drainage
- Drainage: Water discharge in channels, pipes, roof drainage
- Surveying: Documentation of elevation differences and terrain
- Civil Engineering: Foundations, slopes, ramps
Gradient: Definition and Units
Gradient is expressed in several ways:
i [%] = (Δh / L) · 100 (percent)
1 : n = L / Δh (ratio notation, e.g., 1:20 = 5%)
Important Distinctions
- Horizontal Distance L: The projection on the horizontal plane (used for gradient calculation)
- Slant Distance s: The actual distance along the inclined surface (s = √(L² + Δh²))
- Inclination Angle α: The angle relative to the horizontal
Typical Grade Values (Practice Examples)
| Application | Gradient i | Percent | Angle α |
|---|---|---|---|
| Flat road (minimal grade) | 0.003 | 0.3% | ≈0.17° |
| Normal road longitudinal grade | 0.02–0.04 | 2–4% | ≈1.1–2.3° |
| Roof drain | 0.05–0.10 | 5–10% | ≈2.9–5.7° |
| Mountain road / Alpine pass | 0.10–0.15 | 10–15% | ≈5.7–8.5° |
| Very steep road | 0.20 | 20% | ≈11.3° |
Drainage: Minimum Grade per DIN 1986
| Pipe Type / Application | Minimum Grade | Typical |
|---|---|---|
| House sewage line (DN 100–150 mm) | 0.5% (1:200) | 1–2% |
| Downspout / Roof drainage | Vertical or ≥2% | 2–5% |
| Road drainage (open channel) | ≥0.3% | 0.5–2% |
| Stormwater line (road) | ≥0.5% (1:200) | 1–3% |
Approximation for Small Angles
For small gradients (i < 0.1, i.e., < 10%), the approximation holds:
This means: A 5% grade (0.05) corresponds approximately to an angle of 0.05 rad ≈ 2.86°. The error is < 1% for i < 0.2.
Slant Distance vs. Horizontal Distance
In surveying it is important to note:
For small i: s ≈ L · (1 + i²/2)
Example: L = 100 m, Δh = 5 m (i = 5%) → s = √(10000 + 25) ≈ 100.12 m (difference: 0.12 m = 0.12%)
• i = 2% or i : 1 = 1:50 (both mean the same)
• Arrows with percent value or ratio
• Elevation annotations (absolute elevations of points)
For precise calculations, elevation data from surveying should be used.
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