Estimate a stable slope angle for granular materials
The angle of repose is the steepest slope (measured from the horizontal) that a loose, granular material can maintain without sliding. It is commonly used as a quick, first-pass indicator for stockpile slopes, conveyor discharge piles, temporary embankments, and material handling design. Typical materials include sand, gravel, crushed rock, grain, coal, and similar bulk solids.
This page calculates the angle of repose in degrees using a simple relationship between slope angle and the tangent function. You can provide the input in either of two equivalent ways:
- Coefficient of friction (μ): a dimensionless friction parameter (often used as a proxy for internal friction in simplified contexts).
- Slope ratio (rise ÷ run): a geometric measurement of the pile face or slope (for example, 3 m rise over 5 m run gives 0.6).
Important: this is an educational and screening-level estimate. Real slope stability can be strongly affected by moisture, cohesion, compaction, particle size distribution, vibration, drainage, and time-dependent effects. Use conservative design practices and follow applicable codes and professional guidance for safety-critical work.
How to use the calculator (μ or rise/run)
- Choose one input method: enter either μ or rise ÷ run. You do not need both.
- Enter a positive value: the calculator requires a number greater than zero.
- Calculate: press Calculate Angle to compute the angle in degrees.
- Interpret: compare the result to typical ranges for similar materials and apply a safety margin for design.
If you enter both fields, the calculator will use the coefficient of friction μ and ignore the slope ratio. This matches the current tool behavior and avoids conflicting inputs.
Formulas used
The calculator uses the arctangent relationship between a slope ratio and an angle. In both cases, the same mathematical form applies:
- From coefficient of friction:
θ = arctan(μ) - From slope ratio:
θ = arctan(rise/run)
To convert from radians to degrees:
θ (degrees) = arctan(value) × 180 / π
Where value is either μ or rise/run. The calculator displays the angle to two decimal places and echoes the effective input value used.
MathML (same formulas, typeset)
Formula: θ = arctan μ
Formula: θ = arctan h / r
Inputs and measurement guidance
Coefficient of friction μ
μ is dimensionless. In this simplified model, μ acts like a tangent of the limiting slope angle. If you have a measured angle of repose from a test, you can back-calculate μ as μ = tan(θ). If you have a friction coefficient from a relevant test method, use that value directly.
Practical note: published μ values vary widely because particle shape, moisture, and compaction change behavior. If you are uncertain, run a conservative and an aggressive scenario to see how sensitive the angle is.
Slope ratio (rise ÷ run)
The slope ratio is the vertical rise divided by the horizontal run. It is sometimes reported as “1V:2H” style notation; that corresponds to a ratio of 1/2 = 0.5. Measure rise and run along a representative cross-section of the pile face. Avoid local irregularities (rills, small slumps, or equipment tracks) if you want an average slope.
Tip: if you only have a slope angle from a clinometer, you can convert it to a ratio with rise/run = tan(θ).
Worked example (realistic, step-by-step)
Example: You measure a stockpile face and find it rises 3 m over a horizontal distance of 5 m.
- Compute the slope ratio:
rise/run = 3/5 = 0.6. - Enter 0.6 in Slope ratio (rise ÷ run) and leave μ blank.
- The calculator computes:
θ = arctan(0.6) × 180/π ≈ 30.96°.
Interpretation: ~31° is a common ballpark for dry sands. For design, you would typically choose a flatter working slope (for example, several degrees lower) to account for variability, disturbance, and wet conditions.
Second example (using μ): If a material has μ = 0.75, then θ = arctan(0.75) × 180/π ≈ 36.87°. This is consistent with many gravels or crushed materials under dry conditions.
Typical angles of repose (approximate)
Actual values depend on particle shape, gradation, moisture, and handling method. The table below is a rough reference for dry, loosely placed materials. Use it to sanity-check results—not as a design standard.
| Material (dry, loose) | Approx. coefficient of friction μ | Approx. angle of repose (degrees) |
|---|---|---|
| Very rounded sand | 0.3–0.4 | 17°–25° |
| Typical dry sand | 0.4–0.6 | 25°–35° |
| Crushed stone / gravel | 0.6–0.8 | 30°–40° |
| Coal (broken) | 0.5–0.7 | 28°–38° |
| Wheat grain | 0.4–0.6 | 25°–35° |
| Angular rock fragments | 0.8–1.0+ | 35°–45°+ |
If your computed angle is far outside these ranges, double-check whether you entered a ratio correctly (rise/run, not run/rise) and confirm that the value is positive.
How to interpret the result
- Lower angles (≈ 20°–30°): more free-flowing materials (rounded grains, smooth particles) that form flatter piles.
- Moderate angles (≈ 30°–45°): common for many sands, gravels, and crushed rock in dry conditions.
- Higher angles (> 45°): may indicate strong interlocking or cohesion (angular fragments, moisture/cementation). These can appear stable but may fail abruptly if conditions change.
For safety-critical slopes, the angle of repose is not a substitute for a full stability analysis (for example, using shear strength parameters, pore pressure, drainage, and a factor of safety). Treat this output as a quick estimate to support early planning, communication, and scenario comparison.
Limitations and assumptions
This calculator intentionally uses a simple model: it converts a single input value into an angle using arctan. That simplicity is useful for quick checks, but it comes with important limitations:
- Dry, uniform behavior assumed: moisture, fines content, cohesion, and segregation can change the effective angle significantly.
- No external loads or vibration: traffic, equipment, earthquakes, blasting, and repeated dumping can reduce stability and trigger sloughing.
- Idealized geometry: real piles are not perfect planes; local steep spots can fail even if the average slope seems acceptable.
- Field vs. lab differences: a μ value from one test setup may not match field placement, compaction, or drainage conditions.
- No code or factor-of-safety check: the output is not a recommended design slope. Apply appropriate safety factors and follow regulations and professional standards.
If failure could cause injury, property damage, or operational disruption, consult a qualified geotechnical professional and use site-specific testing and analysis.
How the angle of repose is measured in practice
Laboratories and field crews measure the angle of repose several ways, and the method you choose can shift the result by a few degrees, so it is worth understanding what your number represents. In the fixed-funnel method, material is poured through a funnel held at a set height until a stable cone forms on a flat base; the cone's height and base radius give the angle through θ = arctan(height ÷ radius). The tilting-box method slowly raises one end of a container of material until the surface just begins to slide, and the tilt angle at that instant is recorded. A revolving-drum approach tumbles material inside a slowly rotating cylinder and reads the dynamic angle of the moving surface.
These methods do not always agree. A poured cone tends to show the "drained" or static angle, while a tilting box captures the angle at the point of incipient motion, and a rotating drum reports a dynamic angle that can be a few degrees lower because the grains are already in motion. Particle size, pour height, base roughness, and even the operator's technique all introduce scatter. For that reason, engineers usually take several readings and report a representative value with a note on the method, the material condition, and the moisture state rather than treating a single measurement as an exact material constant. When you enter a coefficient of friction or a measured slope into this calculator, you are effectively feeding one of these observed slope ratios back through the arctangent to express it as an angle.
Angle of repose questions people ask
What is the angle of repose?
The angle of repose is the steepest angle (measured from horizontal) at which a pile of loose granular material can remain stable without sliding. It depends on particle friction, shape, moisture, and how the pile is formed.
Do I enter both μ and slope ratio?
No. Enter either the coefficient of friction (μ) or the slope ratio (rise ÷ run). If both are entered, this calculator uses μ and ignores the slope ratio.
What formula does the calculator use?
The calculator uses θ = arctan(μ) or θ = arctan(rise/run), then converts to degrees: θ(deg) = arctan(value) × 180/π.
Does moisture change the angle of repose?
Yes. A little moisture adds capillary cohesion that can raise the angle, letting damp sand stand steeper than dry sand. Too much water saturates the material, removes that cohesion, and can make it flow at a much lower angle. Always note the moisture condition when you record an angle of repose.
Sources: the relationship θ = arctan(μ), where the tangent of the repose angle equals the coefficient of friction for a cohesionless material, is the standard granular-mechanics result described in soil-mechanics texts and summarized by the angle-of-repose literature. Typical material angles follow bulk-solids handling references such as the Engineering ToolBox angle-of-repose table. This tool is a first-pass estimate and is not a substitute for geotechnical slope-stability analysis.
Slopekeeper Mini-Game
Guide a stream of grains to sculpt a stable pile. Keep both slopes just under the critical angle as materials shift and surprises roll in — the closer you ride the edge, the higher your score climbs.
Run Complete
Tip: Keep both slopes within 90–100% of the critical angle to earn stability bonuses. Drag or tap to move the chute; use ← → keys on desktop.