Angle Of Repose Calculator

What the angle of repose calculator does

Pour sand, grain, or gravel onto a flat surface and it forms a cone. The slope of that cone, the steepest angle the loose material can hold without sliding, is the angle of repose. This calculator finds it from the height of the heap and the radius of its base, and also returns the coefficient of static friction that goes with it.

Below is what the angle of repose means, the equations behind it, how it relates to friction, and a worked example.

How to use it

1. Enter the height of the heap, measured from the base to the peak.
2. Enter the radius of the heap's circular base.
3. Press Calculate for the angle of repose and the coefficient of static friction, or Reset to clear it.

What the angle of repose is

The angle of repose is the steepest slope at which a pile of loose, granular material stays put. Build the pile any steeper and the grains on the surface start to tumble down until the slope settles back to that angle. It is a property you can see in any heap of dry sand, a pile of coffee beans, or a mound of gravel, all of which settle to a characteristic cone.

The reason a heap has a limiting slope at all is friction between the grains. As long as friction can hold a surface grain against the pull of gravity along the slope, the grain stays; once the slope is steep enough that gravity wins, it slides. The angle of repose is exactly the slope where those two are in balance, which is why it is the same for a given material no matter how big the pile.

The equations it uses

The heap is a cone, so its slope is set by its height h and base radius r through the tangent. The angle of repose θ is:

θ = arctan( h ÷ r )

The same ratio gives the coefficient of static friction between the grains, since at the angle of repose the friction is on the verge of being overcome:

μ = tan(θ) = h ÷ r

So a taller, narrower heap means a steeper angle and a higher friction coefficient, while a low, spread-out heap means a shallow angle and a slippery material.

The link to friction

The neat result that the coefficient of static friction equals the tangent of the angle of repose comes straight from balancing forces on a grain at the surface. Gravity pulls the grain down the slope, and friction resists it. At the steepest stable slope, the friction is at its limit, and working through the balance shows that the ratio of the two, which is the friction coefficient, is exactly the tangent of the slope angle.

This makes the angle of repose a simple, practical way to measure how grippy a granular material is. Rather than testing friction directly, you can just pour a heap, measure its height and base, and read off the friction coefficient. The same balance is why a slope at the angle of repose is on the edge of sliding, the principle behind both deliberate flow in hoppers and accidental landslides.

What the angle depends on

The angle of repose is not quite a fixed constant for a material, because it depends on the nature of the grains. Rough, angular, or interlocking particles hold a steeper slope than smooth, rounded ones, since they grip each other better. Finer or more irregular material tends to pile higher, while smooth spheres slump to a shallow cone.

Moisture matters too. A little water between grains adds cohesion and lets the slope stand steeper, which is why damp sand builds a sandcastle while dry sand will not. Dry granular materials typically settle somewhere around the low-to-mid thirties of degrees, but the exact figure is best measured for the specific material, which is what this calculator lets you do from a real heap.

Units and precision

You can enter the height and radius in millimetres, centimetres, metres, inches, feet, or yards, and the calculator converts internally, so only their ratio matters. The angle of repose is reported in degrees, with other angle units available, and the friction coefficient is a pure number. Results carry several figures, finer than a poured heap can really be measured, so treat the angle as a sound estimate.

A worked example

Suppose a pile of sand stands 0.3 metres high with a base radius of 0.5 metres.

The angle of repose is arctan(0.3 ÷ 0.5) = arctan(0.6) ≈ 31 degrees, and the coefficient of static friction is 0.3 ÷ 0.5 = 0.6. That sits right in the range expected for dry sand, which typically rests at around 30 to 35 degrees.

Questions people ask

What is the angle of repose?

It is the steepest slope a pile of loose granular material can hold without grains sliding down. Beyond it, the surface flows until the slope settles back to that angle.

How do you calculate it?

From a heap's height and base radius, the angle is arctan(height ÷ radius). The coefficient of static friction equals the tangent of that angle, which is the same height-to-radius ratio.

How is it related to friction?

At the angle of repose, friction between grains is just balancing gravity along the slope. Working out that balance shows the friction coefficient equals the tangent of the angle.

What changes the angle of repose?

Particle shape, size, roughness, and moisture. Angular or damp material holds a steeper slope; smooth, dry, rounded grains settle to a shallower one.

References

A quick note on where the physics comes from. The angle of repose as arctan of a heap's height over its base radius, and its link to the coefficient of static friction, are standard granular mechanics, described in the Wikipedia article on the angle of repose. The SI units follow the US National Institute of Standards and Technology.

1. Wikipedia, Angle of repose. https://en.wikipedia.org/wiki/Angle_of_repose
2. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811