Angle of Repose Calculator

What Is the Angle of Repose?

The angle of repose is the steepest angle, measured from the horizontal, at which a pile of granular material can rest without the particles sliding down the slope. It arises from the balance between the component of gravity pulling particles downhill and the inter-particle friction resisting motion. When the slope angle equals the angle of repose, the material is in a state of limiting equilibrium: any steeper and particles will slide.

It is widely used in civil engineering (embankment and dam design), geotechnical engineering (slope stability), process engineering (silo and hopper design), and geology (natural slope prediction).

The Formula: phi = arctan(mu)

For a cohesionless granular material, the angle of repose (phi) equals the arctangent of the coefficient of static friction (mu) between particles:

phi = arctan(mu)

This comes directly from equilibrium conditions on an inclined plane. At the critical angle, the driving force (mg sin phi) equals the maximum frictional resistance (mu x mg cos phi). Dividing both sides gives tan(phi) = mu, so phi = arctan(mu).

Rearranged: mu = tan(phi). You can therefore measure the angle of a stable pile and read off the effective inter-particle friction coefficient.

From Pile Dimensions

If you form a conical pile and measure its height (h) and the base diameter (d), the slope angle from horizontal is:

phi = arctan(2h / d)

The factor of 2 arises because the radius (half of d) is the horizontal run to the pile apex. For example, a pile 1.2 m high with a base diameter of 4.0 m has a slope ratio of 2 x 1.2 / 4.0 = 0.60, giving phi = arctan(0.60) = 30.96 degrees.

Typical Values for Common Materials

These values are approximate and vary with particle size, shape, moisture content, and compaction. Always test the actual material when designing slopes, hoppers, or storage facilities.

Slope Ratio Interpretation

Engineers often express slope steepness as a ratio of horizontal run to vertical rise (H:V). An angle of 30 degrees corresponds to a slope of about 1.73:1 (run:rise) or, alternatively, a 1:0.58 (rise:run) gradient. Embankment design typically specifies the slope in this H:V form, and the angle of repose sets the minimum safe gradient for the fill material used.

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Sources and method: Coulomb, C.A. (1776). Essai sur une application des regles des maximis et minimis a quelques problemes de statique (classical derivation of the friction-angle relationship). Brandt, D.A. & Warner, J.C. (2009). Metallurgy Fundamentals (material friction values). Angle-of-repose values from geotechnical engineering reference data (Das, B.M., Principles of Geotechnical Engineering, various editions).

This calculator provides indicative values for cohesionless granular materials. Cohesive soils, very fine powders, or materials affected by humidity may behave differently. Consult a geotechnical engineer for design-critical applications.