The angle of repose calculator above handles the geometry of a conical heap of loose material. In one direction it takes a height and a base you have measured and returns the angle those imply. In the other it takes an angle you supply and returns the height, volume, mass and face length that follow from it. It does not predict, assess or certify anything about how a heap will behave.
Arb Digital builds free physics calculators, and this one comes with a warning attached that is more important than the arithmetic. Stockpile and silo failures kill people. Material that looks solid can be a crust over a cavity, and a face that has stood for weeks can run in seconds when something is drawn from underneath it. Nothing on this page tells you a heap is safe, and no calculation can.
What This Angle of Repose Calculator Does
The angle of repose is defined as the steepest angle to the horizontal at which a heap of loose granular material comes to rest when it is poured freely onto a flat surface. It is a descriptive measurement of a particular material in a particular state, taken at a particular moment. It is not a material constant and it is not a design value.
This tool deliberately takes that angle as an input rather than looking it up, and there is no material table anywhere on this page. The reason is that the number genuinely moves. The same sand can sit at noticeably different angles wet and dry; a small amount of moisture forms liquid bridges between grains and steepens the heap, while more moisture lubricates and flattens it. Particle size, particle shape, angularity, how far the material fell as it was placed, and whether it has been driven over all change it too.
What the tool computes is the geometry that an angle implies. A freely poured heap approximates a cone, so height is the base radius multiplied by the tangent of the angle, volume is a third of the base area times the height, and mass is volume times bulk density. Those relationships are exact for a cone and approximate for anything real.
How to Use It
- Decide which direction you are working in. Angle mode turns two measurements into an angle. Geometry mode turns an angle you already have into a shape.
- Measure the base across the toe, not across the shoulder. The toe of a heap spreads further than people estimate, and using the wrong diameter changes the volume by the square of the error.
- Get the repose angle from your own material. Pour a sample freely onto a flat plate and measure the heap you actually get, in the condition the material is actually in.
- Use a bulk density, not a solid density. Loose material is mostly voids. Weighing a filled container of known volume takes a minute and removes the largest source of error in the mass figure.
- Treat every output as indicative. Real heaps are not cones and ground is not flat.
The Formula: How the Geometry Is Calculated
For a right circular cone of base radius r and apex height h, the face makes an angle θ with the horizontal where tan θ = h ÷ r. Rearranged, h = r tan θ. The volume is V = πr²h ÷ 3, the slant length of the face is l = √(r² + h²), and the mass is the volume multiplied by the bulk density of the material.
Work the defaults in geometry mode. A base diameter of 20 m gives r = 10 m. At an angle of 35 degrees, tan 35° = 0.700208, so h = 7.002 m. The volume is π × 100 × 7.002 ÷ 3 = 733.3 m³, and at a bulk density of 1,600 kg/m³ that is 1,173,000 kg, or about 1,173 tonnes. The slant length is √(100 + 49.03) = 12.21 m. In angle mode, a measured height of 7 m on the same 20 m base returns arctan(7 ÷ 10) = 34.99 degrees, which closes the loop.
The angle of repose itself is a measured quantity, and the University of Cambridge DoITPoMS teaching page on the angle of repose covers what it means and how it is determined for granular materials. Take the figure for your own material from your own measurement, not from a published example.
Why a Heap Is Not a Static Object
The most dangerous assumption anyone makes about a pile of loose material is that it is a solid. It is not. It is a very large number of grains held in a fragile arrangement of contact forces, and that arrangement changes on its own.
Heaps slump as vibration and settlement rearrange the grains. They crust, when moisture, freezing or fines bind the surface into a shell that feels like ground but is a skin over loose or absent material. They bridge, when material arches across a void above a feeder, leaving a cavity invisible from the surface. And they run, when drawing from beneath removes the support for a face that then arrives all at once.
Penn State Extension's guidance on the hazards of flowing grain describes what these mechanisms do to a person: flowing grain behaves like quicksand, and a standing adult can be trapped past the knees in seconds and buried in well under a minute. Crusted material collapses when someone walks onto it in order to break it up, dropping them into the cavity underneath. Those are grain examples, but the mechanisms are general to loose material.
The Mine Safety and Health Administration's safety alert on preventing stockpile accidents covers the same hazards on aggregate and surge piles, including cavities formed by bridging over a feeder and material compacted or frozen at the surface. Read it before you go near a working pile.
The practical rules that follow are simple and absolute. Nobody stands on a heap. Nobody stands at or works at the toe of a heap. Nobody enters a bin, silo or bunker to free material. Draw points, feeders and reclaim tunnels are treated as engulfment hazards. This calculator has no opinion on any of that, and neither should the number it produces.
Why the Angle Changes and Why Tables Mislead
Published angle-of-repose figures are measurements of a specific sample under specific conditions. Measure the same material twice, once dry and once damp, and watch the answer move.
Moisture works in both directions. A little water forms capillary bridges between grains, adds cohesion and steepens the heap. Enough water fills the pores, removes the surface tension effect and reduces the friction between grains, and the heap flattens. Between those two states the same material can present as two different materials.
Particle shape matters as much as size. Angular crushed material interlocks and stands steeper than rounded material of the same grading, and a wide range of particle sizes tends to stand steeper than a uniform one because fines fill the gaps. Segregation during placement then leaves different parts of the heap behaving differently.
How the heap was made matters too. Material dropped from a height compacts differently from material placed gently, and a heap built by a conveyor is not the same heap as one built by a loader even with the same material in it.
If you want one figure to hold on to: at roughly 34 degrees, dry sand poured freely into a heap is often cited as an indicative value for that one material in that one state. Do not carry it across to your material, to wet sand, or to sand that has been compacted, and do not use it as a design input.
Where Engineering Takes Over
There is a hard boundary between what this page does and what real work requires. Bins, silos, hoppers, bunkers and reclaim systems are engineered structures, designed under their own standards using flow properties measured in a shear tester rather than a poured heap. Wall friction, arching, ratholing, mass flow versus funnel flow and the loads a discharging silo imposes on its own walls are all part of that design, and none of them can be inferred from a repose angle.
Slopes and excavations are the same story. The stability of ground is a geotechnical question that depends on shear strength, pore water pressure, layering, loading and drainage, and it is assessed by a qualified engineer against the applicable code. An angle of repose describes a freely poured heap of loose grains and is not a slope stability criterion.
Where this calculator genuinely earns its place is in quantity work: estimating the volume and mass of a heap you have measured, or the footprint a given quantity will need. For that side of the job the cone calculator handles the pure geometry, the cubic yard calculator and gravel calculator handle material quantities, and the material weight calculator converts a volume into a mass.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating the angle of repose as a material constant — it is a measurement of one sample in one state, and moisture, grading, shape and handling all move it.
- Reading a repose angle as a stability criterion — it describes where freely poured grains come to rest. It says nothing about whether a slope or a face will hold, which is a geotechnical assessment.
- Using solid density instead of bulk density — loose material is largely voids, so using the density of the grains overstates the mass of a heap substantially.
- Measuring the base at the shoulder — the toe spreads further than it looks, and because volume goes with the square of the radius a small error here becomes a large error in tonnes.
- Assuming the surface tells you what is underneath — crusting and bridging both produce a surface that looks continuous over a cavity, and walking on it is how people die.
Related Free Tools From Arb Digital
For the pure geometry of a cone use the cone calculator, and for quantities and deliveries use the cubic yard calculator, the gravel calculator and the material weight calculator. If you are working with an angle rather than a heap, the slope calculator converts between degrees, gradients and percentages, and the inclined plane calculator and friction force calculator cover the rigid-body physics of a block on a slope, which is a different problem from a heap of grains. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the steepest angle to the horizontal at which a heap of loose granular material comes to rest when poured freely onto a flat surface. It is a measurement of one material in one condition at one moment, not a fixed property of the material.
Because the angle of repose is not reliable enough to tabulate for use. Moisture content, particle size and shape, compaction and the way a heap was built all move it, so a figure copied from a table is a figure for someone else's sample rather than for your material.
Pour a sample of the material freely onto a flat plate from a fixed low height and let it form a heap, then measure the height and the base diameter of that heap and take the arctangent of height divided by base radius. Repeat it in the condition the material will actually be in.
No, and nothing on this page should be read that way. It is a geometry calculation. Whether a heap or a slope will hold is a geotechnical assessment that depends on shear strength, moisture, loading, layering and drainage, and it has to be made by a qualified engineer.
Small amounts of water form capillary bridges between grains, adding cohesion and letting the heap stand steeper. More water fills the pore space, removes that surface tension effect and lubricates the grains, so the heap flattens. The same material can behave very differently at two moisture contents.
Bridging is when material arches across a void, typically above a feeder or draw point, leaving a hidden cavity. Crusting is when moisture, freezing or fines bind the surface into a shell over loose or absent material. Both produce a surface that looks solid and is not, and both have killed people who walked on them.
No. Silo, bin and hopper design is engineered work carried out under its own standards, using flow properties measured in a shear tester along with wall friction, arching and discharge loads. A poured-heap angle is not an input to that process and cannot substitute for it.
Bulk density is the mass of material per unit volume of heap, including the air in the voids between grains. Solid density is the density of the grains themselves. For loose granular material the bulk figure is much lower, and using the solid one badly overstates the mass of a heap.
This tool is provided for educational and estimating use only. It performs conical geometry from figures you supply and makes no assessment of stability, safety or fitness for any purpose. Stockpiles, slopes, silos, bins and bunkers can fail without warning and have caused fatalities; assessment and design of any of them is work for a qualified geotechnical engineer, carried out under the applicable standards and the requirements of your workplace safety regulator.
