Angle of Repose Calculator

What is the angle of repose?

The angle of repose is the steepest angle at which a pile of loose granular material can stand without sliding or collapsing. If you slowly pour sand, gravel, grain, or similar particles onto a flat surface, the sides of the pile form a characteristic slope. That slope, measured from the horizontal, is the angle of repose.

This property reflects how easily particles slide past one another. Materials that are smooth, round, or lubricated by water tend to form flatter piles with a low angle of repose. Materials with rough, angular, or interlocking grains can support much steeper slopes before they fail.

Engineers, geologists, and operators use the angle of repose as a quick indicator of slope stability and flow behavior. It can help with tasks such as designing stockpiles and hoppers, judging whether a soil or aggregate slope is potentially unstable, or comparing how two materials will behave when piled or poured.

Angle of repose formulas

There are two closely related ways to describe the angle of repose mathematically:

1. Using the coefficient of friction between particles.
2. Using the geometric slope ratio (rise divided by run).

From coefficient of friction

The internal friction between grains is often expressed as a dimensionless coefficient of friction, μ. Under simple conditions, the angle of repose θ (in radians) is related to μ by:

In standard calculator notation this is usually written as:

θ = arctan(μ)

If you know μ, the angle of repose in degrees is:

θ (degrees) = arctan(μ) × 180 / π

From slope ratio (rise ÷ run)

You can also measure the slope directly as a rise/run ratio. If a pile or slope rises a vertical height h over a horizontal distance r, then the slope ratio is:

slope = h / r

The corresponding angle of repose is given by the same tangent relationship:

So if you measure the geometry of the slope, you can compute the angle of repose directly from the rise/run ratio.

How to use the Angle of Repose Calculator

The calculator accepts either the coefficient of friction μ or the slope ratio (rise ÷ run). You only need to enter one value to compute the angle.

1. Choose your input type:
   • If you know or can estimate the material’s coefficient of friction, enter that in the “Coefficient of friction μ” field. Typical dry granular materials often have μ roughly between about 0.3 and 0.8.
   • If you measured a pile or slope, compute its rise/run ratio and enter that in the “Slope ratio (rise ÷ run)” field. For example, a 1 m rise over 2 m run corresponds to a ratio of 0.5.
2. Enter only one value if possible: For clarity, fill in either μ or the slope ratio. If you do enter both, the calculator will use the coefficient of friction and ignore the slope ratio.
3. Click “Calculate Angle”: The tool computes the angle of repose in degrees using the arctangent of the chosen input.
4. Review and copy the result: Compare the resulting angle with typical values for similar materials, or use it to assess whether a planned or existing slope is unusually steep.

Interpreting your results

The output is the theoretical or estimated angle at which a pile of your material just begins to slide. Interpreting this value correctly is important for safe design and practical decision making.

• Lower angles (for example, 20°–30°): The material is relatively free-flowing. Sand with rounded grains or dry grain often falls in this range. Piles will be wide and flat.
• Moderate angles (about 30°–45°): Common for many soils, gravels, and crushed rock. These materials can form fairly steep but still free-draining slopes.
• Higher angles (greater than about 45°): Indicate strong interlocking or cohesion, such as angular rock fragments or moist, cohesive soils. These slopes may appear stable but can fail suddenly if conditions change.

In practice, designers rarely use the exact angle of repose as a working slope angle in safety-critical structures. Instead, they apply a safety margin and design slopes that are flatter than the measured or calculated angle of repose to account for variability and uncertainty.

Worked example

Suppose you are evaluating a dry sand stockpile at an aggregate yard. You measure the pile and find that it rises 3 m over a horizontal distance of 5 m from the toe to the crest. You want to estimate the angle of repose for this sand.

1. Compute the slope ratio: rise/run = 3 / 5 = 0.6.
2. Enter the value: In the calculator, leave the coefficient of friction field blank and enter 0.6 in the slope ratio field.
3. Click “Calculate Angle”: The tool computes θ = arctan(0.6) ≈ 30.96°.
4. Interpret the result: An angle of roughly 31° is typical for many dry sands. If your site guidelines recommend keeping stockpile slopes below, say, 28° for additional safety, you might decide to regrade the pile to a flatter slope.

As another example, assume your lab tests show that a gravel material has an internal friction coefficient of μ = 0.75 under conditions similar to field compaction. Using the calculator with μ = 0.75 gives:

θ = arctan(0.75) ≈ 36.9°

If you are planning a temporary gravel embankment, you might initially estimate that slopes around 30°–33° could be reasonable, subject to more detailed geotechnical checks and any relevant codes or standards.

Typical angles of repose by material

Actual values depend strongly on particle shape, gradation, moisture, and handling, but the following approximate ranges are commonly cited for dry, loosely placed materials.

Use these values only as rough guidance. Whenever possible, measure the behavior of your specific material under representative conditions.

Limitations and assumptions

The angle of repose calculator is a simplified tool and relies on important assumptions. Keep these in mind before using any output for design or safety decisions:

• Dry, homogeneous material: The formulas assume relatively dry, uniform material. Moisture, layering, cementation, or inclusions (e.g., trash, organics) can significantly alter the true angle of repose.
• No external loads or vibration: Real slopes are affected by traffic, equipment, earthquakes, blasting, and other dynamic effects. Vibration tends to reduce stability, so actual safe angles may be lower than calculated.
• Idealized geometry: The calculations assume a simple, planar slope. Irregular shapes, benches, and local overhangs can create zones of higher stress and earlier failure.
• Laboratory vs. field conditions: Values of μ or angle measured in the lab may not match the field if compaction, moisture, or grading differ. Treat lab-derived angles as indicative, not absolute.
• No regulatory or code check: The tool does not apply building codes, mine safety regulations, or geotechnical design standards. In regulated environments, you must follow applicable codes and, if needed, consult a qualified engineer or geoscientist.
• Safety factors required: The calculated angle is not a recommended design angle. Appropriate safety factors, drainage considerations, and long-term degradation should be evaluated in a full stability assessment.

Because slope stability can be safety-critical, use this calculator as an educational and preliminary estimation aid only. For important projects or where failure could cause harm or significant loss, a detailed geotechnical analysis and professional review are essential.