The angle of repose is calculated using a simple trigonometric formula: measure the height of a conical pile of material and the radius of its base, then take the inverse tangent of height divided by radius. Written out, the formula is: Angle = tan⁻¹(h / r), where h is the pile height and r is the radius. Most granular materials produce an angle between 25° and 40°, though moisture, particle shape, and grain size can shift that number significantly.
The Core Formula
Picture a pile of sand or gravel sitting on a flat surface. It naturally forms a cone. If you slice that cone straight down the middle, the cross-section is a triangle. The angle at the base of that triangle, where the sloped surface meets the ground, is the angle of repose.
To calculate it, you only need two measurements. First, measure the height (h) of the pile from the flat surface to the peak. Second, measure the radius (r) of the pile’s base, which is the distance from the center to the outer edge. Then plug those into the formula:
Angle of repose = tan⁻¹(h / r)
This comes from basic right-triangle trigonometry (the “TOA” in SOHCAHTOA). The tangent of an angle equals the opposite side divided by the adjacent side. Here, the opposite side is the pile’s height and the adjacent side is the radius. Taking the inverse tangent (sometimes written as arctan) converts that ratio back into degrees. Any scientific calculator or phone calculator in scientific mode has this function.
Step-by-Step Measurement
The easiest way to measure angle of repose at home or on a job site is the fixed-base funnel method. Pour your material slowly through a funnel held above a flat surface, letting the pile build naturally without disturbing it. The funnel should be close enough that falling material doesn’t scatter on impact, but far enough that the pile doesn’t touch the funnel opening.
Once the pile stops growing outward, measure its height with a ruler or straightedge placed vertically at the peak. Then measure the diameter of the base and divide by two to get the radius. For accuracy, take measurements from at least two or three different piles and average the results. Inconsistent pouring speed or vibration can skew individual readings.
For a quick worked example: if your pile is 12 cm tall and the base radius is 18 cm, the calculation is tan⁻¹(12 / 18) = tan⁻¹(0.667) = 33.7°. That would indicate a material with good flow properties.
What the Number Tells You
The angle of repose is a practical indicator of how easily a granular material flows. Lower angles mean the material spreads out readily and flows freely. Higher angles mean particles lock together and resist movement. A widely used classification scale, originally developed by the materials scientist Ralph Carr, breaks it down like this:
- 25°–30°: Excellent flow
- 31°–35°: Good flow
- 36°–40°: Fair flow
- 41°–45°: Passable flow
- 46°–55°: Poor flow
- 56°–65°: Very poor flow
- 66°+: Extremely poor flow
This scale is used across industries, from pharmaceutical powder manufacturing to agricultural storage. If you’re designing a hopper, chute, or storage bin, knowing where your material falls on this scale determines the minimum slope your equipment needs to keep things moving.
Static vs. Dynamic Angle of Repose
The formula above gives you the static angle of repose, which is what most people need. But there are actually two distinct versions worth understanding, especially if you’re working with material that moves continuously.
The static angle has an upper and lower value. As you gradually add grains to a pile, the slope steepens until it hits an upper angle, at which point an avalanche occurs. After the slide, the surface settles at a lower angle. The upper angle is essentially the steepest slope a pile can hold before collapsing, while the lower angle is what remains after a collapse.
The dynamic angle of repose describes the surface inclination when grains are flowing continuously down a slope, like inside a rotating drum. Research using rotating drums half-filled with uniform sediment has shown that the dynamic angle is roughly equal to the average of the upper and lower static angles. The gap between the upper and lower angles grows as particle size increases, meaning coarser materials have more dramatic avalanche events.
Factors That Change the Angle
Moisture Content
Wet materials pile steeper than dry ones. Water creates surface tension between particles, which acts like a weak glue holding grains together. Research on rice grains found the angle of repose increased steadily as moisture content rose from 10% to 30%, climbing from roughly 22° to 29°. The friction between particles and surrounding surfaces also increases with moisture, which is why damp sand holds a sandcastle shape while dry sand collapses flat. If you need a consistent measurement, dry your material to a known moisture level before testing, or at minimum record the moisture content alongside your angle reading.
Particle Shape and Size
Angular, jagged particles interlock with each other and produce steeper piles. Round, smooth particles roll past one another and settle at lower angles. Of all the ways to describe particle shape, roundness has the strongest correlation with angle of repose. Crushed rock, for instance, will have a notably higher angle than river-tumbled gravel of the same mineral.
Particle size matters too, but in a direction that surprises some people. Smaller particles generally produce higher angles of repose, not lower ones. Fine powders have more surface area relative to their weight, which means friction and electrostatic forces play a bigger role. Very fine powders can become cohesive enough to pile at angles above 50°, which is why flour and cocoa powder clump and resist flowing through narrow openings.
Using the Angle for Slope Stability
In civil engineering, the angle of repose feeds directly into slope safety calculations. For a simple slope made of cohesionless material (like clean sand or gravel, with no clay holding it together and no water pressure), the factor of safety is calculated as:
Factor of safety = tan(friction angle) / tan(slope angle)
The friction angle in this context is closely related to the angle of repose. If the slope angle equals the friction angle, the factor of safety is 1.0, meaning the slope is right at the edge of stability. Design standards from the U.S. Army Corps of Engineers require factors of safety well above 1.0 for any permanent structure, typically 1.3 to 1.5 depending on the consequences of failure. So if your material has an angle of repose of 35°, you’d want to build your embankment or stockpile at a slope shallower than 35° to maintain a safety margin.
This simplified calculation only applies to dry, cohesionless soils. Real-world slopes with clay content, groundwater, or mixed materials require more complex analysis. But for estimating stockpile dimensions, designing storage berms, or figuring out how much floor space a gravel pile will occupy, the basic angle of repose calculation is all you need.
Standardized Testing Methods
If your measurement needs to meet industry standards, ISO 4324 specifies a standardized funnel method for powders and granules. The standard controls variables like funnel height, base surface material, and environmental conditions (temperature and humidity) to ensure repeatable results. Pharmaceutical applications reference the USP chapter on powder flow, which uses the same Carr flowability scale listed above. For most practical purposes outside a laboratory, the funnel-and-ruler method described earlier gives results accurate enough for engineering estimates and material handling decisions.