Angle of Repose Calculator
This calculator works out the angle of repose, the steepest slope a pile of loose granular material can hold before it slides, using two methods you can switch between with the tabs above. The first method uses the coefficient of static friction (mu) between particles, either typed in directly or chosen from a preset for common materials such as dry sand, gravel, wheat, coal, wood chips, dry soil or limestone powder, and applies the formula phi = arctan(mu). The second method works from a real pile you have measured, taking the height and base diameter of a cone and using phi = arctan(2h / d), which also back-calculates the effective friction coefficient. Either way, you get the angle of repose in degrees and radians, the friction coefficient, and the slope expressed as a rise-to-run ratio, alongside a full working breakdown and a worked example using dry sand at mu = 0.577. You can adjust the number of decimal places shown for extra precision. Use it to check embankment and stockpile slopes, size hoppers and silos, or sense-check a measured pile against typical values for the material. The results are indicative for cohesionless, free-flowing materials only; wet, cohesive or very fine powders can hold steeper angles than predicted, so treat outputs as a planning guide and consult a geotechnical engineer for design-critical work.
1. Material
2. Output options
Full Breakdown
Worked Example
What Is the Angle of Repose?
The angle of repose is the steepest angle, measured from the horizontal, at which a pile of granular material can rest without the particles sliding down the slope. It arises from the balance between the component of gravity pulling particles downhill and the inter-particle friction resisting motion. When the slope angle equals the angle of repose, the material is in a state of limiting equilibrium: any steeper and particles will slide.
It is widely used in civil engineering (embankment and dam design), geotechnical engineering (slope stability), process engineering (silo and hopper design), and geology (natural slope prediction).
The Formula: phi = arctan(mu)
For a cohesionless granular material, the angle of repose (phi) equals the arctangent of the coefficient of static friction (mu) between particles:
phi = arctan(mu)
This comes directly from equilibrium conditions on an inclined plane. At the critical angle, the driving force (mg sin phi) equals the maximum frictional resistance (mu x mg cos phi). Dividing both sides gives tan(phi) = mu, so phi = arctan(mu).
Rearranged: mu = tan(phi). You can therefore measure the angle of a stable pile and read off the effective inter-particle friction coefficient.
From Pile Dimensions
If you form a conical pile and measure its height (h) and the base diameter (d), the slope angle from horizontal is:
phi = arctan(2h / d)
The factor of 2 arises because the radius (half of d) is the horizontal run to the pile apex. For example, a pile 1.2 m high with a base diameter of 4.0 m has a slope ratio of 2 x 1.2 / 4.0 = 0.60, giving phi = arctan(0.60) = 30.96 degrees.
Typical Values for Common Materials
| Material | Coefficient of Friction (mu) | Angle of Repose (approx) |
|---|---|---|
| Dry sand | 0.53 to 0.67 | 28 to 34 deg |
| Wet sand | 0.84 to 1.00 | 40 to 45 deg |
| Gravel (rounded) | 0.65 to 0.84 | 33 to 40 deg |
| Gravel (angular) | 0.84 to 1.19 | 40 to 50 deg |
| Coal | 0.70 to 0.84 | 35 to 40 deg |
| Wheat / grain | 0.49 to 0.54 | 26 to 28 deg |
| Limestone powder | 0.40 to 0.49 | 22 to 26 deg |
| Wood chips | 0.84 to 1.00 | 40 to 45 deg |
| Dry soil (loose) | 0.58 to 0.70 | 30 to 35 deg |
These values are approximate and vary with particle size, shape, moisture content, and compaction. Always test the actual material when designing slopes, hoppers, or storage facilities.
Slope Ratio Interpretation
Engineers often express slope steepness as a ratio of horizontal run to vertical rise (H:V). An angle of 30 degrees corresponds to a slope of about 1.73:1 (run:rise) or, alternatively, a 1:0.58 (rise:run) gradient. Embankment design typically specifies the slope in this H:V form, and the angle of repose sets the minimum safe gradient for the fill material used.
Related Calculators
- Maths and Stats Calculators: full hub for geometry, trigonometry, and statistics tools.
- Angle of Elevation Calculator: find the elevation angle from height and distance.
- Angle of Depression Calculator: calculate depression angles for surveying and optics.
- Slope Calculator: find gradient, rise, and run for any slope.
- Friction Calculator: calculate friction force from normal force and friction coefficient.
Sources and method: Coulomb, C.A. (1776). Essai sur une application des regles des maximis et minimis a quelques problemes de statique (classical derivation of the friction-angle relationship). Brandt, D.A. & Warner, J.C. (2009). Metallurgy Fundamentals (material friction values). Angle-of-repose values from geotechnical engineering reference data (Das, B.M., Principles of Geotechnical Engineering, various editions).
This calculator provides indicative values for cohesionless granular materials. Cohesive soils, very fine powders, or materials affected by humidity may behave differently. Consult a geotechnical engineer for design-critical applications.