What this calculator does
Pour sand into a pile and it settles at a characteristic angle. Add more and the pile grows wider but not steeper, because any surface steeper than that angle simply slides. That limit is the angle of repose.
It comes straight from friction: the angle at which the tangent equals the static friction coefficient. Everything from grain silo design to the stability of an embankment depends on knowing it for the material involved.
The formula
Take the arctangent of the static friction coefficient. The result is the angle at which the gravity component down the slope exactly balances the friction available to resist it.
| Term | Meaning |
|---|---|
| Angle of repose | The steepest stable angle for a loose granular material. |
| Static friction coefficient (μs) | The friction ratio between grains at rest, which determines the angle. |
| Slope ratio | The same angle expressed as rise over run, which is often how earthworks are specified. |
The inputs explained
| Field | What to enter |
|---|---|
| Static friction coefficient (μs) | The static friction coefficient of the material against itself. Dry sand is typically around 0.6 to 0.7. |
When to use it
Designing a stockpile or hopper
Storage capacity depends on the cone shape the material naturally forms, which follows from this angle.
Assessing an embankment
A batter cut steeper than the angle of repose will not hold without engineered support.
Estimating a material property
Measuring the angle of a poured pile works backwards to the friction coefficient, which is hard to measure directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What angle does each friction coefficient produce?
A range of coefficients from smooth to very grippy.
| Friction coefficient | Angle of repose (θr) | Maximum stable slope (rise : run) |
|---|---|---|
| 0.3 | 16.70 ° | 0.300 : 1 |
| 0.6 | 30.96 ° | 0.600 : 1 |
| 0.8 | 38.66 ° | 0.800 : 1 |
| 1 | 45.00 ° | 1.000 : 1 |
Questions
Why does a sand pile stop getting steeper?
Because any grain landing on a slope steeper than the repose angle has more gravity pulling it down the slope than friction holding it, so it rolls until the surface flattens back to that angle.
Does moisture change the angle?
Considerably. A little moisture adds cohesion between grains and allows much steeper slopes, which is why damp sand builds castles and dry sand does not. Too much water removes the effect entirely.
Does grain size matter?
Yes. Angular, rough grains interlock and hold steeper slopes than smooth rounded ones. Grain shape often matters more than grain size.
Is this the same as the slope of a safe excavation?
No, and treating it that way is dangerous. The angle of repose applies to loose poured material. Excavation batters depend on soil type, moisture, loading and groundwater, and must follow the relevant engineering and safety requirements.
For friction on a solid object rather than loose material, see the friction force calculator. For slope expressed as a grade, see the slope percentage calculator.