Choose a method and solve for a variable
Sample inputs and expected outputs
| Scenario | Inputs | Angle (deg) | Grade (%) | mu |
|---|---|---|---|---|
| Cone geometry | Height 18 cm, Radius 25 cm | 35.75 | 72.00 | 0.72 |
| Slope | Rise 12 m, Run 20 m | 30.96 | 60.00 | 0.60 |
| Percent grade | Grade 50% | 26.57 | 50.00 | 0.50 |
| Friction coefficient | mu 0.65 | 33.02 | 64.99 | 0.65 |
| Reverse solve | Angle 40°, Cone radius 10 cm | 40.00 | 83.91 | 0.84 |
Values are rounded for display. Your results depend on precision settings.
Key relationships
- tan(θ) = h / r for a conical pile (height h, radius r).
- tan(θ) = rise / run for a slope profile.
- Grade(%) = 100 × tan(θ), so θ = arctan(Grade/100).
- μ ≈ tan(θ) when using an ideal friction equivalence.
Angle of repose varies with moisture, particle shape, vibration, and packing density.
Steps
- Select a method that matches your measurements.
- Choose the variable you want to solve for.
- Set units and precision, then enter required values.
- Press Calculate to view results above the form.
- Use CSV or PDF buttons to export your result.
1) What angle of repose means
Angle of repose is the steepest stable slope a granular pile can hold without sliding. It is measured between the pile surface and the horizontal base. Higher values usually indicate more interlocking, rougher particles, stronger cohesion, or lower vibration during placement.
2) Geometry behind the cone method
If you pour material into a cone, the basic model uses tan(θ)=h/r, where h is the vertical height and r is the base radius. This calculator accepts radius or diameter, supports several length units, and returns both degrees and radians.
3) Slope, grade, and friction equivalence
In field work, you may measure a slope using rise and run, so tan(θ)=rise/run. Road-style grade is Grade(%)=100×tan(θ). A common engineering shortcut links repose to friction through μ≈tan(θ), giving a quick way to compare stability across materials.
4) Typical values you may see
Ranges depend on moisture and handling, but many dry materials often fall between about 25° and 45°. Fine dry sand frequently lands near the low-to-mid 30° range, while coarse gravel can trend higher. Grain products such as wheat may be closer to the upper 20s to low 30s. Treat these as rough benchmarks.
5) How measurement setup affects results
Drop height, pouring rate, and vibration can shift the final slope. A higher drop can compact particles and change packing density. Vibration often lowers the slope by allowing grains to settle. For consistent data, keep the same funnel height, base plate size, and fill procedure across tests.
6) Moisture and cohesion effects
A small amount of moisture can increase cohesion and temporarily raise the stable angle, while too much moisture may cause slumping or flow. Powders with electrostatic effects may also show higher stability than expected from particle size alone. Record humidity, moisture content, and time since handling.
7) Why engineers track it
Angle of repose is used in hopper design, belt transfer points, stockpile planning, and safety checks for bulk storage. It helps estimate how far a pile spreads, the space required for storage, and the likelihood of avalanching. The calculator’s reverse-solve modes support quick planning from limited measurements.
8) Tips for better calculator inputs
Use the cone method for lab-style piles, the slope method for measured profiles, and the grade or friction modes for quick conversions. Avoid mixing radius and diameter unless you intend it. If your angle approaches 90°, recheck inputs because very steep slopes usually indicate a measurement or unit issue.
1) Is angle of repose the same as angle of friction?
They are related but not identical. Angle of repose describes a free pile surface, while friction angle is a material property used in models. This tool offers μ≈tan(θ) as a practical equivalence.
2) Which method should I use: cone or slope?
Use cone when you can measure a poured pile’s height and base radius. Use slope when you have rise and run from a profile or a photo-based measurement. Both compute the same angle when the geometry matches.
3) Why does my result change between tests?
Small changes in moisture, particle size distribution, drop height, and vibration can shift packing and stability. Keep the setup consistent and average several trials to reduce random variation.
4) Can I enter diameter instead of radius?
Yes. If diameter is provided, the calculator automatically uses r=diameter/2. If both are entered, radius takes priority only when it is present and valid.
5) What does grade (%) mean here?
Grade is a slope expressed as a percent: 100×(rise/run). For example, a 50% grade means 0.5 units up for every 1 unit across, which corresponds to about 26.6°.
6) What if I only know the coefficient μ?
Choose the friction method and solve for angle. The calculator uses θ=arctan(μ) to provide degrees, radians, and the equivalent grade. This is a convenient estimate for quick comparisons.
7) Is this suitable for safety-critical design?
Use it for screening and planning, then validate with material-specific testing and standards for final design. Real bulk behavior depends on consolidation, wall friction, moisture migration, and handling conditions.