Angle of Repose Calculator — Granular Material Stability
The angle of repose is the steepest slope at which a heap of granular material remains stable without sliding. Select a material preset or enter the coefficient of static friction to find the critical angle, slope gradient, and percentage slope.
Material
Maximum stable slope angle for this granular material at rest
- 1
arctan(μ) in radians
arctan(0.55) = 0.502843The angle of repose θ satisfies tan θ = μ (the coefficient of static friction). - 2
Angle of repose
0.502843 × (180 ÷ π) = 28.8
How does this calculator work?
Angle of repose θ = arctan(μ), where μ is the coefficient of static friction. Dry sand (μ ≈ 0.55) reposes at about 29°; gravel (μ ≈ 0.84) at about 40°. Friction equals the slope gradient at the repose angle: μ = rise / run. Preset μ values are engineering averages (2020); measure site samples for geotechnical work.
Formula
How this is calculated
When granular material (sand, gravel, grain) is piled up, the particles interlock and resist sliding through friction. The angle of repose θ is the steepest slope the heap can sustain — it is defined purely by the static friction coefficient μ between particles: θ = arctan(μ). Conversely, if you measure the angle of a natural stable pile, the friction coefficient is μ = tan(θ).
Physically, on a slope at angle θ, gravity pulls each surface particle with a force component along the slope of mg·sinθ (driving it to slide) and presses it into the slope with mg·cosθ (providing the normal force for friction). Maximum static friction is μ·mg·cosθ. At the angle of repose, these exactly balance: mg·sinθ = μ·mg·cosθ, which simplifies to tan θ = μ. The heap is stable at any angle below θ and collapses above it.
The values in the presets are approximate engineering averages (circa 2020). Real angles of repose depend on moisture content, particle shape, size distribution, and compaction state — wet rounded grains differ substantially from angular dry ones. For civil engineering and geotechnical work, always measure site-specific samples rather than relying solely on published averages.
Frequently asked questions
At low to moderate moisture, surface tension in the thin water film between particles creates capillary bridges — extra cohesion that lets wet material stand at steeper angles than dry material. At very high moisture (fully saturated) the cohesion disappears and the material may flow like a liquid, reducing the effective repose angle dramatically.
It determines safe embankment and trench slopes, hopper and silo wall angles, and stockpile dimensions. Any designed slope must be kept below the angle of repose (plus a safety factor). Retaining walls, geotextiles, or soil nailing are used when the required slope exceeds the natural repose angle for the material.
For dry, cohesionless granular materials (pure sand or gravel) the angle of repose equals the internal friction angle φ in the Mohr-Coulomb failure criterion. For cohesive soils (silt, clay) there is an additional cohesion term, and the observable repose angle is higher than the internal friction angle alone.
Also known as
TG we-Calculate Editorial Team. (2026). Angle of Repose Calculator — Granular Material Stability [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angle-of-repose-calculator
TG we-Calculate Editorial Team. "Angle of Repose Calculator — Granular Material Stability." TG we-Calculate. 2026. https://we-calculate.com/calculator/angle-of-repose-calculator.
TG we-Calculate Editorial Team, "Angle of Repose Calculator — Granular Material Stability," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angle-of-repose-calculator
@misc{wecalculate_angle_of_repose_calculator, title = {Angle of Repose Calculator — Granular Material Stability}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angle-of-repose-calculator}}, year = {2026}, note = {TG we-Calculate} }
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