About

The angle of repose θ is the steepest angle a granular material can sustain before sliding. It is defined as θ = arctan(hr), where h is the pile height and r is the base radius. Errors in estimating this angle lead to silo wall failure, conveyor spillage, or slope collapse. The value depends on particle shape, moisture content, and surface friction. This calculator solves for any unknown variable given two knowns and computes the factor of safety FoS = tan(φ)tan(θ) against the material's internal friction angle φ.

Results assume dry, cohesionless, uniformly graded particles at standard gravity. Cohesive soils, wet granules, or vibrated conditions will deviate. The presets use published mean values from ASTM D6128 and DIN 18126 test standards. Pro tip: always measure repose angle at the ambient moisture your process actually operates at. Laboratory dry values can underestimate field angles by 3 - 8°.

Formulas

The angle of repose is derived from the geometry of a conical pile. Given the pile height and the base radius, the angle at the base equals:

θ = arctan ( hr )

Where θ = angle of repose (°), h = vertical height of the pile (m), and r = horizontal radius of the base (m). When the base diameter d is known instead, r = d2.

Inverse solutions allow computing unknown dimensions:

h = r ⋅ tan(θ)

r = htan(θ)

The factor of safety against sliding is computed by comparing the material's internal friction angle φ to the actual slope angle θ:

FoS = tan(φ)tan(θ)

Where FoS ≥ 1.0 indicates the slope is at or below the material's natural limit. Values below 1.0 predict slope failure. Engineering practice requires FoS ≥ 1.5 for permanent structures per Eurocode 7 and AASHTO LRFD.

The conical pile volume is calculated as:

V = 13 ⋅ π ⋅ r² ⋅ h

Where V = volume (m³). Multiplying by bulk density ρ gives estimated pile mass: m = ρ ⋅ V.

Reference Data

Material	Angle of Repose (°)	Internal Friction Angle φ (°)	Bulk Density (kg/m³)	Particle Shape
Dry Sand (fine)	34	30 - 36	1500	Sub-rounded
Dry Sand (coarse)	37	33 - 40	1600	Sub-angular
Gravel	40	35 - 45	1800	Angular
Wheat	27	25 - 30	770	Ellipsoidal
Corn (shelled)	27	24 - 30	720	Rounded
Rice	36	33 - 38	690	Elongated
Cement (Portland)	40	38 - 42	1500	Angular powder
Coal (bituminous)	38	35 - 40	830	Angular
Limestone (crushed)	38	35 - 42	1540	Angular
Iron Ore	35	32 - 38	2500	Sub-angular
Sugar (granulated)	35	32 - 37	850	Cubic
Salt (table)	32	30 - 35	1200	Cubic
Flour	45	40 - 50	590	Cohesive powder
Alumina	35	32 - 38	960	Sub-angular
Wood Chips	45	40 - 50	350	Irregular
Fly Ash	42	38 - 45	700	Spherical
Glass Beads	22	20 - 25	1500	Spherical
Sawdust	38	35 - 42	210	Fibrous
Soil (dry clay)	40	35 - 45	1100	Platy
Topsoil (loam)	35	30 - 38	1300	Mixed
Snow (dry)	38	30 - 45	100	Dendritic
Plastic Pellets	25	22 - 28	600	Spherical

Frequently Asked Questions

Moisture introduces capillary cohesion between particles, increasing the apparent angle of repose by 3-12° depending on saturation level. Fine materials like flour or fly ash are most sensitive. The calculator's presets assume dry conditions. For moist materials, measure experimentally or add a moisture correction factor to the dry value.

The angle of repose θ is a bulk, unconfined surface phenomenon measured by pouring material into a pile. The internal friction angle φ is a shear-strength parameter obtained from triaxial or direct-shear laboratory tests under controlled confining pressure. For dry, cohesionless, uniformly graded particles, θ ≈ φ. For cohesive or poorly graded materials, they diverge significantly. The factor of safety calculation in this tool uses both values.

Angular particles interlock mechanically, resisting sliding and producing higher repose angles (38-50°). Spherical particles like glass beads roll freely, yielding low angles (20-25°). Elongated or platy particles (rice, clay flakes) stack irregularly and produce intermediate to high values. Shape effects dominate over size for most granular systems.

Only as a first-order approximation for infinite, dry, cohesionless slopes. Real slope stability requires methods like Bishop's simplified method or Janbu's method, which account for pore water pressure, soil layering, surcharge loads, and circular failure surfaces. Use the factor of safety output here as a screening tool, not a design basis.

The ASTM C1444 fixed-funnel method pours material through a funnel onto a flat surface and measures the cone dimensions. The tilting-box method (ASTM D6393) slowly inclines a box until material slides. Results vary by ±2-4° between methods. The funnel method tends to give slightly higher values. Report the method used alongside the angle.

A FoS of 1.0 means the slope angle equals the material's critical angle. The pile is at the verge of failure. Any perturbation (vibration, wind, added material) will trigger sliding. Engineering codes require minimum FoS of 1.3 for temporary works and 1.5 for permanent structures. Never design at FoS = 1.0.