Angle of Repose Calculator

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Our angle of repose calculator can help you find a measure of the tendency of a granular material to flow or help you with your designs in case you know the property of the material and you need to model its behavior.

Keep reading this short but exhaustive article to learn:

What is the angle of repose? Definition and scope of this quantity.
How to calculate the angle of repose formula: direct and inverse calculations.
And example of calculations of the angle of repose: a pile of soil.

And much more! What are you waiting for? start pouring!

What is the angle of repose: definition and example

The angle of repose has a valid definition only for granular materials (think of sand, salt, flour...). In these cases, when a material is poured from above on a horizontal surface, it will eventually reach a point where its slope will not increase anymore. The angle formed with the horizontal plane is called angle of repose, and is an intrinsic characteristic of the material and a few other parameters.

The angle of repose is a direct measure of the aptness of a material to flow: the lower the angle, the more the material flows (think about it: water has an angle of repose of $0$ : it flows so well!), while for higher angles we witness materials that oppose to flowing. The granulometry of a material is one of the core elements that affect the angle of repose: the coarser the material, the higher the angle. Check how to find volume with density and mass for structural load calculations.

Once the heap of material reaches the associated angle of repose (all heaps start from angle $0\degree$ if built on a flat surface), the inclination of the slope won't increase anymore, and any additional material will flow on the side of the pile.

In the next section, we'll learn the formula to calculate the angle of repose: it's surprisingly easy!

How to calculate the angle of repose: formula and explanation

The calculations of the angle of repose rely heavily on the modelization of friction, in particular static friction, as we are interested in the behavior of the heap on the edge of flowing.

Definting the

\theta_\mathrm{r} = \arctan(\mu_\mathrm{s})

where:

$\theta_\mathrm{r}$ — Angle of repose;
$\mu_\mathrm{s}$ — Static friction coefficient.
$\arctan$ — Inverse tangent function.

This formula quickly tells you what is the angle of repose for a given material. Notice how for lower static friction coefficients, the angle of repose is similarly low: this feature corresponds to materials that flow easily, where the number density of individual particles directly influence inter-particle friction.

If you don't know the coefficient of static friction, or if the conditions are different than usual (for example, if the material is wet), you can use the relationship between the measurements of the heap to calculate the angle of repose:

\tan(\theta) = \frac{h}{r}=\frac{2\cdot h}{d}

where:

$r$ — Radius of the heap.
$d$ — Diameter of the heap ( $d=2\cdot r$ ).
$h$ — Height of the heap.

To find this result, we used the property we met in the tangent calculator and the right triangle trigonometry calculator.

Notice that by considering the rules of trigonometry in right triangles and the formulas of the inclined plane calculator, we can find another expression for the friction coefficient in our setup:

\mu = \frac{m\cdot g\cdot \sin(\theta)}{m\cdot g \cdot \cos(\theta)}=\frac{\sin(\theta)}{\cos(\theta)}

where the coefficient is the ratio between the horizontal component of the force due to friction and the vertical component of the gravitational force (the normal force) acting on each grain of the material. Unlike structured cubic cell lattices, granular friction relies on random contact points.

🙋 In the last formulas we switched from $\mu_\mathrm{r}$ to $\mu$ , without index: this is because calculating the angle of repose of a heap starting by measurements on the heap itself calculates the coefficient of static friction for specific circumstances, while starting from the tabulated values of $\mu_\mathrm{r}$ gives a forecasted value regardless of other factors.

An example: how to calculate the angle of repose of soil

Let's see how to calculate the angle of repose formula for soil. The friction coefficient for this material is variable, but we'll choose a reference value of $$\mu_\mathrm{s} =

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