Angle Of Repose Interactive Calculator

If you’re designing a conveyor discharge, a grain silo hopper, or a mine waste stockpile, you need to know one thing: the maximum angle you can pile material before it starts to slip. This Angle of Repose Calculator lets you work that out—whether you’ve got pile geometry, lab friction data, or just field measurements. It’s relevant anywhere bulk piles matter, such as mining, agriculture, powder-handling, and earthworks. A low safety margin here means your pile could give way, so it pays to check the numbers. You'll also find the governing equations, a worked coal stockpile example, technical notes on how granular materials behave, and an FAQ that covers practical differences between static/dynamic angle, the real effects of moisture, and measurement approaches.

What is the Angle of Repose?

Angle of repose is just the steepest slope a loose pile of granular stuff—think sand, grain, rock fragments, or powder—will naturally hold before grains start sliding. It’s a direct result of how much friction the particles have with each other.

Simple Explanation

If you dump dry sand onto a surface, it forms a cone and stops steepening at a particular angle—that angle is the angle of repose. Round, smooth particles make flat piles, while rough, gritty particles stack up steeper. Add a little moisture and the angle can increase, because water helps the grains stick together for a while.

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Angle of Repose Diagram

Angle of Repose Calculator

How to Use This Calculator

1. Pick what you want to solve: angle from geometry, pile dimensions, friction coefficient, volume, or slope stability.
2. Put in your numbers—height, base, angle, coefficient, density, whatever applies.
3. Double-check your units: metres for dimensions, degrees for angles, kg/m³ for density.
4. Hit Calculate and check the output.

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Equations & Formulas

To get angle of repose from a pile’s dimensions, use the formula below.

For friction coefficient, relate it directly to the repose angle as follows:

To get the pile’s volume assuming it’s conical, use the equation here.

If you have the angle and either height or radius, you can solve for the missing dimension:

This is the fast way to estimate a basic factor of safety on a granular slope:

Simple Example

A sand pile has a height of 3 m and a base radius of 4 m. What is the angle of repose?

• Height (h) = 3 m, Base radius (r) = 4 m
• θ = arctan(3 / 4) = arctan(0.75)
• θ = 36.87°
• Friction coefficient: μ = tan(36.87°) = 0.75

Theory & Practical Applications

Fundamental Physics of Granular Materials

Angle of repose comes from gravity pulling grains down the slope, resisted by friction between the particles. As you add material, grains keep sliding off the pile’s edge until the slope settles at an angle where gravity pulling down the surface matches the grip—or friction—between the grains below. If you pour more or disturb the pile, it’ll settle back to roughly the same angle every time. This is the basic parameter engineers care about for anything involving loose bulk materials.

Materials like sand or gravel don’t stick together like clay or mud; it’s mostly about particle contacts—the rougher and sharper the particles, the steeper the pile can get before it collapses. The math is simple: angle of repose is arctangent of the friction coefficient. Smooth, round particles fall somewhere around 23–28°. Sharp, angled, or irregular shapes can be much steeper, sometimes exceeding 40°. Small changes in shape and size mix will shift the angle, so lab numbers are a guideline, not a guarantee.

Engineers should keep in mind: angle of repose is not a single absolute value. If you measure it on a stopped pile, it’ll be several degrees higher than what you get with the material in motion (like in a rotating drum or flowing through a chute). Expect the static angle to be 5–10° above the dynamic value. For anything involving flow, use the dynamic angle; for slope and pile stability, use static. Some powders—especially pharmaceutical types—can swing even wider due to humidity or static electricity. If you just use the textbook number, you risk overestimating or underestimating your actual pile stability or flow.

Industrial Applications Across Sectors

Mining and Aggregate Handling: In mining, stockpiles and waste dumps are sized and shaped using repose angle as a key input, then cross-checked by geotechnical methods. As an example, copper ore at 37° may stand at 39° right after dumping but settles back once fines wash down after rain. For conveyors, design around the dynamic angle (which can be several degrees less) to make sure the load falls where you want it and doesn’t spill.

Agricultural Storage: Grain bins rely on a good repose angle estimate for capacity and load calculations. Wheat, for example, sits near 28°, but above 14% moisture it can steepen above 32–35°. Soybeans have an even lower angle, so require wider bins for the same height. As grain is discharged and a cone shape forms, the pressure on the walls changes, and the repose angle helps you model that so you don’t overload the structure.

Pharmaceutical Manufacturing: Hopper flows for tableting are limited by the repose angle. Lactose monohydrate might need a cone angle of 50–55° in the hopper to ensure it flows—because its repose angle is 38–42°, anything less and you’ll get bridging. Powders with angles over 50° are basically unworkable unless you add flow agents to lower the angle, otherwise the powder just sits there.

Construction and Earthwork: For sand and aggregate on construction sites, you need to know the angle to keep stockpiles out of work areas and prevent a slip onto workers or machines. Dry sand is usually 30–35°, but add fine clay and it holds at 38–42°. Truck drivers should not drive onto a slope even close to the repose angle—a good rule of thumb is no steeper than (repose angle)/1.5, giving you some safety margin.

Advanced Engineering Considerations

If you want to predict how fast material flows out of hoppers and silos, you need more than just the angle of repose—the Beverloo equation adds particle size and friction factors. The “k” factor in Beverloo is bigger for angular, high-friction materials, which reduces the discharge rate with the same orifice. If you size chutes or feeders on angle of repose alone, you might get choked flow or bridging, and need to increase the opening or change the hopper shape.

In geotechnical slope analysis, angle of repose is a starting guess for sandy or gravel slopes. The quick factor of safety calculation is just tan(repose angle)/tan(actual slope). Real slopes are complicated by things like water, layering, or vibration, but if your number is below 1, your slope will likely slip. For a temporary embankment with a repose angle of 38° and built at 35°, safety factor is 1.25—just enough for something short-term, not a permanent unrestricted use.

Stockpiles almost always segregate by size as you dump them. Larger chunks roll farther and end up at the bottom edge, while fine material collects near the top. That means your pile is not exactly one angle everywhere. Expect a few degrees steeper at the base and a couple degrees lower at the tip, especially for a large pile and variable size material. If consistent material is vital—such as for cement kilns—use stacking devices to control where material lands and reduce segregation.

Worked Example: Coal Stockpile Design

If you’re putting up a coal stockpile for a 50,000 tonne plant inventory, and your lab tests say the dry static angle is 36.5°, here’s the approach. Assume a wedge-shaped pile on an 80 m long area, and find base width and pile height that fit the total volume, then check the safety margin for both dry and after-rain (when the angle can jump due to moisture absorption).

Given Data:

• Total mass: M = 50,000 tonnes = 50,000,000 kg
• Coal bulk density: ρ = 850 kg/m³ (typical for stockpiled sub-bituminous coal)
• Stockpile length: L = 80 m
• Dry repose angle: θ_dry = 36.5°
• Wet repose angle: θ_wet = 39.2°
• Required safety factor: FS ≥ 1.5

Step 1: Calculate Required Volume

V = M / ρ = 50,000,000 kg / 850 kg/m³ = 58,823.5 m³

Step 2: Model Pile Geometry

Assume wedge geometry, so V = ½ × base width × height × length. Relate height and width with tan(angle) = h/(w/2).

Solve for width and height using these relationships. Here the answer is about 63 m wide and 23 m high to fit the dry pile into 80 m length, but you’ll see in the next step the safety factor’s tight.

Step 3: Calculate Pile Height

h = (w/2) × tan(36.5°) = (63.04/2) × 0.7400 = 23.32 m

Step 4: Verify Volume

V_check = (1/2) × 63.04 × 23.32 × 80 = 58,822 m³ —

Step 5: Stability Analysis Under Dry Conditions

The slope angle of the as-built pile equals the repose angle (36.5°), so:

FS_dry = tan(36.5°) / tan(36.5°) = 1.00

You’re right at the stability limit. To leave a safety margin (say, FS 1.5), you’ll need a shallower slope, around 26.3°.

Step 6: Adjusted Design for Required Safety Factor

Using θ = 26.3°, you have to go wider and lower. New numbers: width ~77 m, height ~19 m.

Step 7: Wet Condition Analysis

If you get a rain and the pile’s angle rises to 39°, your safety factor improves, but unless you designed for the lower (dry) value, you won’t know your margin. In this example, the lower angle design still meets the minimum after rain. Pile designers typically add berms, account for traffic compaction, and don’t rely on theoretical maximums—real piles almost always shift due to rain, vibration, or settlement, so it’s better to leave slack in the design.

Practical Implications: Actual coal piles get compressed by loader traffic (which can locally raise the slope), and you have to include practical features like drainage to keep water out. That’s why standards call for safety factors—not just building to the angle the lab measured with dry, loose material. For similar calculations, you can use other calculators in the linked engineering tools library.

Frequently Asked Questions