Angle of Repose Calculator (from pile height and base radius)

Understanding the angle of repose calculator — concept and principle

The angle of repose is the steepest angle at which a pile of loose granular material, such as sand, gravel, grain or powder, stays stable. This calculator finds it from the height and base radius of a poured cone, and also gives the slope, side length and pile volume.

A cone 1.2 m high on a 2 m radius has sides at arctan 0.6 ≈ 31°, typical of dry sand.

How it is calculated

Angle of repose θ = arctan(h ÷ r), where h is the pile height and r the base radius.
Slope = h ÷ r, also shown as a percentage.
Slant length of the side = √(h² + r²).
Cone volume V = π r² h ÷ 3.

The principle

Particles on the surface of a pile are held by friction and interlocking with the particles below. If the slope gets steeper than a critical angle, the gravitational pull down the slope exceeds the friction and grains slide until the angle drops back. For a cohesionless material, tan θ is roughly equal to the internal friction coefficient. Particle shape, size distribution, moisture and vibration all change the angle.

Worked example

Default inputs: height 1.2 m, base radius 2 m.

Slope = 1.2 ÷ 2 = 0.6, or 60%.
θ = arctan 0.6 ≈ 30.96°, shown as 31°.
Base diameter = 4 m; slant length = √(1.2² + 2²) ≈ 2.33 m.
Volume = π × 2² × 1.2 ÷ 3 ≈ 5.03 m³.

Each point is the calculator re-run with only “Base radius r (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Base radius r (m)	Angle of repose (°)
0.5	67.4
1	50.2
1.5	38.7
2	31
2.5	25.6
3	21.8
4	16.7

How to read the result

Dry sand typically settles at roughly 30–35°, though values vary with grain shape and moisture. Use the measured angle to size stockpiles, hoppers and storage, or to compare powders' flowability: lower angles generally mean freer flow. The angle describes the loose material tested, not compacted fill.

Common mistakes

Measuring a pile that was compacted or disturbed.
Pouring from a large height, which changes the pile shape.
Using the angle of repose as a design slope for engineered embankments.
Mixing radius and diameter.

Keep in mind: The angle of repose is not a substitute for geotechnical slope design. Real earthworks and storage structures need local codes and a qualified engineer.

Frequently asked questions

What is the angle of repose of sand?

Dry sand commonly lies around 30–35°, depending on grain shape and moisture.

How do I measure the angle of repose?

Pour the material from a low, fixed height onto a rough flat base, measure the cone's height and base radius, and take arctan(h ÷ r).

Why does wet sand stand steeper?

A little water creates capillary cohesion between grains, allowing steeper slopes.

Sources: ASTM C1444, Standard Test Method for Measuring the Angle of Repose of Free-Flowing Mold Powders · Nedderman, R. M., Statics and Kinematics of Granular Materials. Cambridge University Press

Checked 9 Nov 2026. Standard geometry of a right circular cone (θ = arctan(h/r), V = πr²h/3); the angle of repose as the static measure of a heap’s side slope is the standard definition used in powder and bulk-solids handling.

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