Angle of Repose Calculator

Determine the angle of repose from stockpile geometry (height & radius). Evaluate material flowability, critical for silos, conveyor belts, and geotechnical analysis. Interactive cone visualization with real-time updates. Verified against ISO 4324:2023 standards.

Horizontal distance from cone center to toe.
Vertical height from base to apex.
? Common materials (adjusted to typical angles):
?️ Dry Sand (r=2.0, h=1.20 → 31.0°)
? Wheat Grain (r=3.0, h=1.35 → 24.2°)
? Portland Cement (r=2.2, h=1.35 → 31.5°)
? Crushed Gravel (r=2.8, h=2.00 → 35.5°)
⚫ Bituminous Coal (r=2.5, h=1.25 → 26.6°)
Client-side processing: All calculations and graphics are performed locally. No data is transmitted or stored.

What is Angle of Repose?

The angle of repose (θ) is the steepest angle at which a granular material remains stable without sliding or collapsing. It is defined by the relationship between the heap height (h) and the base radius (r): θ = arctan(h / r). This fundamental property governs bulk solid behaviour in stockpiles, hoppers, silos, and natural slopes like talus cones.

θ = arctan( h / r )   →   tanθ = height / radius

From geotechnical engineering to pharmaceutical powder processing, the angle of repose predicts flowability, segregation potential, and wall friction. Low angles (≤30°) indicate excellent flow, while high angles (>45°) suggest cohesive, poor-flowing materials. This calculator implements the fixed-base cone method described in ISO 4324:2023 (Surface active agents – Determination of angle of repose).

Engineering & Industrial Significance

  • Hopper & Silo Design: Ensures reliable discharge; steeper hopper walls required for high-angle materials.
  • Conveyor Belt Capacity: Maximum incline angle for bulk transport without rollback.
  • Stockpile Volume Estimation: Accurate repose angle refines resource inventory (mining, aggregates).
  • Landslide & Avalanche Hazard: Natural granular deposits (scree, volcanic tephra) follow repose thresholds.
  • 3D Printing Powders: Flowability affects layer spreading and part density.

Reference Table: Typical Repose Angles for Common Materials

Material Angle of Repose (deg) Flowability
Dry sand (fine) 30° – 35° Good
Wheat / corn 23° – 28° Excellent
Portland cement 30° – 40° Fair
Crushed gravel (3/4”) 35° – 45° Fair-Poor
Wet clay / topsoil 40° – 50° Poor
Coal (bituminous) 25° – 35° Good
Limestone powder 38° – 45° Fair
Metal powders (iron) 35° – 42° Fair
注：*Actual angle is significantly influenced by particle sphericity, surface moisture, and particle size distribution. The values in the table above are typical laboratory dry-state values; on-site sampling and measurement are recommended. For cohesive powders (θ > 45°), calibration using a shear cell tester (ASTM D6773-16) is advised.*
Source: Jenike & Johanson flowability indices, ASTM D6393-14, and "Bulk Solids Handling" by Schulze (2021). Values are indicative; actual repose depends on moisture, particle shape, and size distribution. The preset examples have been calibrated to match these typical ranges.

How to Use This Calculator

  1. Measure or estimate the base radius of your conical stockpile (distance from center to toe).
  2. Measure the vertical height from ground to apex.
  3. Enter values (any consistent length unit) – the angle is dimensionless.
  4. Click "Compute Angle & Draw" – view the angle, flowability class, and interactive diagram.
  5. Use material presets to explore typical values for sand, cement, grain, etc. (corrected to match reference table).
Case Study: Hopper Design for Cement Plant

A cement plant measured stockpile radius = 4.2 m, height = 2.3 m → θ = arctan(2.3/4.2) ≈ 28.7°. This indicates fair-to-good flowability. Engineers specified a 60° conical hopper with carbon steel liner to prevent arching. Using the computed angle, mass flow was achieved with vibration assist only during extreme humidity. The calculator allowed rapid what-if scenarios. This matches the methodology in "Guidelines for the Design of Silos" (DIN 1055-6).

Theoretical Derivation & Limitations

The angle of repose is directly related to the internal friction angle (φ) for coarse, cohesionless materials: θ ≈ φ. For fine, cohesive powders, the measured angle depends on the method (fixed funnel, tilting box, or heap formation). Our calculator uses the geometric heap method (ISO 4324), which is most representative for industrial stockpiles. Limitation: assumes ideal conical shape, no moisture, and homogeneous material. Real-world deviations may require correction factors. For materials with θ > 55°, the heap becomes unstable and the calculator will still provide the mathematical angle but physical interpretation requires caution.

? Difference Between Dynamic and Static Angle of Repose
This calculator is based on the static pile method (fixed cone). For certain powders (e.g., wet granulated materials), the dynamic angle of repose (rotating drum method) may be 5°–10° higher than the static value. The dynamic angle better reflects material flow behavior during conveying and mixing. For cohesive powders (θ > 45°) or critical process design, cross‑validation using a shear cell tester (ASTM D6773‑16) or the rotating drum method is recommended. Reference: "Powder Technology: Fundamentals of Particles, Powder Beds, and Particle Generation" (H. Masuda, 2020).

Relationship with Euler's Mechanics & Granular Flow

The stability condition at repose is given by the balance of gravitational and frictional forces along the slope: μ = tanθ, where μ is the coefficient of interparticle friction. For bulk solids, the effective friction angle is obtained from shear testers; however, the angle of repose serves as a rapid field indicator. The simple tangent relation (h/r) remains universally valid for conical heaps.

Frequently Asked Questions

Angles range from 0° to about 55°. Most granular materials fall between 25° and 45°. Very free-flowing materials (peas, plastic pellets) may be ≤25°, while sticky powders exceed 50°.

Yes, as long as height and radius share the same unit, the ratio and angle remain correct. Use any consistent length system (mm, cm, feet).

Moisture content, particle shape, compaction, and measurement method (dynamic vs. static) cause variation. Our calculator gives a theoretical geometric baseline; always verify with on-site testing for critical applications.

It helps design stable embankments, estimate debris flow runout, and classify granular soils (USCS). Combined with internal friction angle, it supports slope stability analysis (e.g., infinite slope model).

The calculation uses double-precision floating point arithmetic (error < 1e-10 degrees). The main uncertainty comes from input measurements. For engineering use, we recommend measuring height and radius with ±2% accuracy.

Knowing the repose angle θ and base radius r, you can calculate heap height h = r·tanθ. Then the conical stockpile volume V = (1/3)πr²h = (1/3)πr³·tanθ. For example, with r = 5 m and θ = 35° (tan ≈ 0.700), h = 3.50 m, volume ≈ 91.6 m³. This is a standard method for inventory estimation in mining and agriculture.
Peer-reviewed references: "Standard Test Method for Angle of Repose of Free-Flowing Moulding Sand" (AFS); Nedderman, R.M. "Statics and Kinematics of Granular Materials" (Cambridge, 2005); ScienceDirect – Angle of Repose. Tool validated by GetZenQuery Tech team (April 2026).