Angle of Repose Calculator
Enter the height and radius of a granular-material heap to find its angle of repose and the equivalent static friction coefficient. Or flip the calculation: enter the friction coefficient to get the angle, or enter the angle directly to convert it. The flowability class updates instantly, and the steps panel shows every stage of the math.
Formula
Worked example
A cone of dry sand has a heap height of 10 cm and a base radius of 14.14 cm. The slope ratio is 10/14.14 = 0.7074. The angle of repose is arctan(0.7074) = 35.26 degrees, placing the sand in the "Fair" flowability class. The equivalent static friction coefficient is tan(35.26 degrees) = 0.7074.
What is the angle of repose?
The angle of repose is the steepest slope at which a heap of loose granular material remains stable without sliding. When material is poured onto a flat surface it forms a cone; the angle between the side of that cone and the horizontal is the angle of repose. A small angle means particles slide easily past each other (free-flowing material), while a large angle means the particles interlock or clump (cohesive material). Engineers use this value to size hoppers, chutes, conveyor inclines, and storage silos, and pharmaceutical scientists use it to characterise powder flow before tableting or capsule filling.
How the angle of repose is calculated
The simplest measurement is the funnel-heap method: pour the material through a fixed-height funnel onto a flat surface, measure the height H and base radius R of the resulting cone, and apply the formula theta = arctan(H/R). Because tan(theta) equals H/R, the result is also the coefficient of static friction between the particles: mu = tan(theta). Equivalently, if the friction coefficient is known from other tests, the angle follows as theta = arctan(mu). This calculator supports all three directions: geometry to angle, angle to friction coefficient, and friction coefficient to angle.
Flowability classification
Several classification systems link angle of repose to practical powder flow. The table used here follows the commonly cited pharmaceutical and bulk-solids scale. Angles up to about 25 degrees indicate excellent free-flowing materials (dry granules, smooth seeds). The 25-35 degree range covers good to fair flow, typical of dry sand and cereal grains. From 35 to 45 degrees materials are described as passable to poor, needing well-designed equipment to avoid bridging. Above 45 degrees materials are cohesive - wet clays, fine sticky powders - and usually require vibration or air-assisted handling to flow reliably. Above about 65 degrees the material is essentially self-supporting and will not flow freely at all.
Factors that affect the angle and practical limitations
The measured angle of repose depends on particle size, shape, surface roughness, moisture content, and the pour height and speed used during the test. Round, smooth, dry particles give low angles; irregular, rough, or moist particles give higher ones. Because of this sensitivity, the angle of repose is a comparative and screening tool rather than a precise design parameter. For reliable hopper design in bulk solids handling, the angle of repose should be supplemented with a shear-cell test (Jenike or ring shear), which measures the actual yield locus of the material and accounts for consolidation stress. The angle of repose also differs from the angle of internal friction used in soil mechanics and silo-load calculations, which is determined by different methods.
Typical angle of repose by material
| Material | Angle of repose (degrees) | Friction coeff. (mu) | Flowability |
|---|---|---|---|
| Corn (shelled) | 21-25 | 0.38-0.47 | Excellent |
| Cement powder (dry) | 20-30 | 0.36-0.58 | Good |
| Wheat | 23-28 | 0.42-0.53 | Good |
| Dry sand (fine) | 30-35 | 0.58-0.70 | Fair |
| Coal (run of mine) | 35-40 | 0.70-0.84 | Passable |
| Gravel (smooth) | 35-40 | 0.70-0.84 | Passable |
| Crushed stone | 40-45 | 0.84-1.00 | Poor |
| Flour (fine) | 40-45 | 0.84-1.00 | Poor |
| Wet sand | 35-45 | 0.70-1.00 | Poor |
| Wet clay | 50-60 | 1.19-1.73 | Very poor |
Approximate values under dry, uncompacted conditions. Moisture, particle size, and shape significantly affect real-world measurements.
Frequently asked questions
What is the angle of repose formula?
The standard formula is theta = arctan(H/R), where H is the vertical height of the granular heap and R is the base radius. This is equivalent to theta = arctan(mu), where mu is the static friction coefficient between the particles. Both forms give the same result because tan(theta) = H/R = mu by definition.
What is a typical angle of repose for sand?
Dry, fine sand typically has an angle of repose between 30 and 35 degrees. Wet sand is steeper, often 35 to 45 degrees, because capillary moisture temporarily binds the particles. Gravel and crushed stone range from 35 to 45 degrees depending on particle shape and angularity.
How do I measure the angle of repose in a lab?
The most common method is the funnel-heap (fixed-cone) technique: pour the material from a fixed height through a narrow funnel onto a flat surface until the pile stops growing. Measure the height of the cone with a ruler and the diameter of the base, then calculate R as half the diameter. Apply theta = arctan(H/R). For better accuracy, repeat three times and average. Other methods include the tilting-box method (tilt a box of material until it slides) and the rotating-drum method, which give slightly different results for the same material.
What does a high angle of repose mean for hopper design?
Hoppers must be steep enough to promote gravity flow. A common rule of thumb is that the hopper half-angle (measured from vertical) should be at least 10 to 15 degrees less than the angle of repose. For a material with an angle of repose of 40 degrees, a mass-flow hopper half-angle of around 25 to 30 degrees is a starting point. For cohesive materials (angle above 45 degrees), a formal Jenike hopper design using the flow factor is strongly recommended to prevent arching at the outlet.
Is angle of repose the same as angle of internal friction?
No, although both describe resistance to flow. The angle of repose is measured by observing the free surface of a poured heap and reflects inter-particle friction and interlocking at the outer surface. The angle of internal friction is determined by a shear-cell or triaxial test under a range of confining stresses and characterises bulk shear strength. For cohesive powders under consolidation, the angle of internal friction is typically larger than the angle of repose and is the correct parameter for silo-load and hopper-design calculations.
Why does moisture affect the angle of repose so much?
A small amount of moisture creates liquid bridges between particles. These capillary bridges add cohesive force that resists sliding, so the pile can stand at a steeper angle. Beyond a threshold moisture level the bridges coalesce and the material becomes saturated, reducing cohesion and causing the angle to drop back toward that of a slurry. This non-linear response means moisture content should always be reported alongside any angle-of-repose measurement.