Angle of Repose Calculator

How to calculate an angle of repose from μ or rise/run

1. Select one representation of the pile: enter either μ or rise ÷ run; both are not required.
2. Use a positive number: a repose slope must be entered as a value greater than zero.
3. Calculate the pile angle: press Calculate Angle to obtain degrees from horizontal.
4. Assess the slope in context: compare it with observations of similar material and allow an appropriate design margin.

When both boxes are populated, this angle-of-repose calculator gives priority to the coefficient of friction μ and does not use the slope ratio. That precedence prevents two competing inputs from producing an ambiguous result.

Angle of repose formulas used by this calculator

This calculator applies the arctangent relationship that converts the pile-face ratio into an angle from horizontal. Whether the supplied value is friction or measured geometry, the computational form is the same:

• From coefficient of friction: θ = arctan(μ)
• From slope ratio: θ = arctan(rise/run)

The angle-of-repose result is converted from radians to degrees as follows:

θ (degrees) = arctan(value) × 180 / π

Here, value is either μ or rise/run. The result is shown to two decimal places, along with the effective friction value used in the calculation.

Angle of repose formulas in MathML

Formula: θ = arctan ⁡ μ

Formula: θ = arctan ⁡ h / r

Angle of repose inputs and field-measurement guidance

Coefficient of friction μ for a granular pile

μ has no units. In the calculator's simplified repose model, it represents the tangent of the limiting slope angle. A measured angle can therefore be converted back to a corresponding μ using μ = tan(θ). If a relevant material test supplies a friction coefficient, enter that number directly.

Practical note: reported μ values can differ greatly as grain shape, moisture, and compaction change. For a preliminary slope comparison, test a range of credible μ values rather than treating one published value as universal.

Measured slope ratio (rise ÷ run)

A slope ratio is vertical rise divided by horizontal run. In familiar “1V:2H” notation, the value is 1/2 = 0.5. Take rise and run from a representative cross-section of the pile face, rather than from a rill, a small slump, or an equipment track if the goal is an average pile slope.

Tip: a clinometer angle can be converted for this input with rise/run = tan(θ).

Worked angle-of-repose example from a measured pile face

Example: A stockpile face rises 3 m over a 5 m horizontal distance.

1. Find the pile's slope ratio: rise/run = 3/5 = 0.6.
2. Enter 0.6 in Slope ratio (rise ÷ run), leaving μ empty.
3. The calculator evaluates θ = arctan(0.6) × 180/π ≈ 30.96°.

Interpretation: a slope near 31° can be a useful preliminary comparison point for dry sand, but it is not a prescribed working or design angle. A flatter slope may be selected to accommodate disturbance, changing moisture, and material variability.

Second angle-of-repose example using μ: With μ = 0.75, the calculation is θ = arctan(0.75) × 180/π ≈ 36.87°. A higher μ produces a steeper calculated limiting slope because its arctangent is larger.

Typical approximate angles of repose by material

Granular angle-of-repose values vary with particle shape, gradation, moisture, and placement. This table is only a broad reference for dry, loosely placed material, useful for checking the scale of an estimate rather than specifying a design slope.

A result far beyond these broad ranges is a reason to verify the entry: the calculator expects rise/run, not run/rise, and requires a positive value. It may also signal that the material condition differs from the dry, loose reference descriptions.

How to interpret an angle of repose result

An angle-of-repose result indicates the modeled steepness of a granular pile face measured above the horizontal; it is not, by itself, a safe engineered slope recommendation.

• Lower angles (≈ 20°–30°): generally correspond to freer-flowing material, such as rounded grains or smoother particles, that settles into flatter piles.
• Moderate angles (≈ 30°–45°): are often encountered with dry sands, gravels, and crushed rock.
• Higher angles (> 45°): can reflect interlocking or cohesion from angular fragments, moisture, or cementation. Such piles can still fail abruptly when conditions change.

For a slope where failure is consequential, an angle of repose does not replace a stability assessment based on shear-strength parameters, pore pressure, drainage, geometry, and a factor of safety. Use this output for early planning, material discussions, and comparison of input scenarios.

Angle of repose limitations and assumptions

This angle-of-repose calculator intentionally reduces the pile to one positive input and converts it with arctan. That makes the result transparent for quick slope checks, while also leaving out conditions that control real granular stability:

• Dry, uniform behavior assumed: moisture, fines, cohesion, and segregation can materially alter the observed repose angle.
• No applied loading or vibration: equipment traffic, blasting, seismic motion, repeated dumping, and vibration can initiate sloughing or reduce stability.
• Idealized pile geometry: an actual stockpile is not a perfect plane, and a local oversteepened area can fail although its average face appears acceptable.
• Test-to-field variation: a μ value measured under one test setup may not represent placement, compaction, drainage, or confinement in the field.
• No code or safety-factor assessment: the output is not a recommended design slope. Apply applicable safety factors, regulations, and professional standards.

If a granular-slope failure could cause injury, damage, or operational loss, obtain site-specific testing and analysis from a qualified geotechnical professional.

How angle of repose is measured for real materials

Angle-of-repose measurements can be made in the laboratory or field, and the chosen method can move the observed result by several degrees. In the fixed-funnel method, material is poured through a funnel at a set height until a cone stabilizes on a flat base; cone height and base radius give θ = arctan(height ÷ radius). In a tilting-box method, one end of a container is raised slowly until the material surface begins to slide, and the tilt at incipient movement is recorded. A revolving-drum test tumbles the material in a rotating cylinder and measures the moving surface's dynamic angle.

Those approaches need not produce identical angle-of-repose values. A poured cone commonly represents a static or drained pile, a tilting box observes onset of motion, and a rotating drum measures material already moving, which can yield a lower dynamic angle. Pour height, particle size, base roughness, and operator technique also introduce scatter. Engineers therefore commonly report several readings together with the test method, material condition, and moisture state rather than treating one observation as an exact constant. Entering μ or a measured rise/run value here simply expresses that observed limiting slope as an arctangent angle.

Angle of repose questions people ask

Sources: the relationship θ = arctan(μ), where the tangent of the repose angle equals the coefficient of friction for a cohesionless material, is the standard granular-mechanics result described in soil-mechanics texts and summarized by the angle-of-repose literature. Typical material angles follow bulk-solids handling references such as the Engineering ToolBox angle-of-repose table. This tool is a first-pass estimate and is not a substitute for geotechnical slope-stability analysis.