Angle Of Repose Calculator - Pile Geometry to Slope Angle

An angle of repose calculator turns a measured granular pile into the slope angle that material will hold when poured onto a flat surface. It takes the pile height and base diameter, applies tan(theta) = h / r, and returns the angle in degrees and radians with the slope ratio, cone volume, and pile mass. Students use it to confirm a textbook angle for sand or gravel, and engineers use it to size hoppers, stockpile pads, and truck loads.

• • Physics and soil mechanics labs: Measure a sand or soil pile poured through a funnel and confirm the angle of repose against the textbook value.
• • Stockpile and aggregate planning: Estimate how much volume a crushed stone or pea gravel stockpile will occupy at the resting angle before siting a pad.
• • Hopper and chute design: Pick a wall slope steeper than the angle of repose so the material slides, or shallower so it stays put.
• • Bulk solids handling and storage: Predict the base footprint of a grain, salt, or fertilizer pile inside a silo or warehouse.

The angle of repose describes a stable, freely poured pile. It is set by inter-grain friction, grain shape and size distribution, and moisture content. The same material poured from a height piles at a lower angle than material placed by hand.

The resting angle is governed by inter-grain friction, and the Friction Factor Calculator covers the head loss and pressure drop that the same friction produces in a pipe.

The calculator takes the vertical pile height and the base diameter, converts the diameter to a radius, and applies tan(theta) = h / r. It reports the angle in degrees and radians, the slope ratio, the cone volume, and the pile mass.

• h (pile height): Vertical distance from the floor to the apex of the pile, in meters.
• d (base diameter): Average diameter of the pile base at the floor, in meters. Twice the radius from the foot of the height line to the edge.
• r (base radius): Half of the base diameter. Equal to d / 2.
• rho (bulk density): Bulk density of the granular material in kilograms per cubic meter. Used to estimate pile mass from the cone volume.
• theta (angle of repose): Slope angle from horizontal to the side of the pile. Common values are 25 to 45 degrees for most granular materials.

The angle and slope ratio come from the same h and r pair, so they always agree. Cone volume uses V = (1/3) pi r^2 h for a right circular cone. The stability class compares the computed angle to the typical range for the chosen preset.

According to the Engineering Toolbox - Angles of Repose, dry sand typically tips at 35 degrees, crushed stone at 30 degrees, with different values for different bulk materials.

The same h and r pair drives the cone volume V = (1/3) pi r^2 h, and the Cone Volume Calculator solves that formula directly so the pile mass and the angle can be checked against each other.

Four ideas cover the physics behind a granular pile. Knowing them helps when a measured pile disagrees with a textbook value.

These four ideas explain why two piles of the same material can sit at different angles.

The angle of repose is set by gravity and inter-grain friction, and the Forces Newton's Laws Calculator works through the weight, normal, and friction force components that produce that balance on a single grain.

The form is organized so geometry inputs sit on the first row and supplementary inputs below. Read the angle from the primary result card, then use the cone volume and pile mass rows to size a stockpile or container.

1. 1 Enter the pile height: Type the vertical distance from the floor to the apex of the pile in meters. The default 1 m matches the example below.
2. 2 Enter the base diameter: Type the average diameter of the pile base in meters. The default 2.4 m is a 1.2 m radius.
3. 3 Add the bulk density: Type the bulk density in kg/m^3. Defaults to dry sand at 1600. A preset in the next row writes a typical value here.
4. 4 Pick a material preset: Choose dry sand, wet sand, pea gravel, crushed stone, dry earth, wheat grain, or loose snow to load a typical bulk density and label the stability class.
5. 5 Read the angle and slope ratio: The angle of repose appears in degrees in the primary result card. The slope ratio and radian value sit below for spreadsheet use.
6. 6 Use the volume and mass rows: Multiply the cone volume by the actual bulk density to scale the result to a real stockpile or container.

A lab pours dry sand through a funnel and measures a 1 m high pile with a 2.4 m base. Enter h = 1, d = 2.4, leave density at 1600, and pick the dry sand preset. The angle card reads 39.81 degrees, just above the textbook 34 but still inside the dry-sand range.

Because the angle itself comes from arctan(h / r), the Arctan Calculator returns the same theta in degrees and radians for any h and r pair, a quick way to spot-check the result card.

The angle of repose is one of the smallest numbers in geotechnical work, yet it controls how a granular pile behaves. A quick numerical answer beats a manual arctangent.

• • Angle in degrees and radians: Read the slope angle in degrees for engineering notes and in radians for spreadsheet trig formulas without a unit conversion step.
• • Slope ratio output: Get the run:rise slope ratio form that civil and geotechnical drawings expect, alongside the degrees form.
• • Cone volume and pile mass: Estimate the conical pile volume and the pile mass from the same h and r inputs, so one measurement gives both the angle and the stockpile size.
• • Material preset for sanity checks: Pick a material to write a typical bulk density and compare the measured pile against the typical angle range for dry sand, wet sand, pea gravel, crushed stone, dry earth, wheat grain, or loose snow.
• • Stability class label: See at a glance whether the computed angle is below, inside, or above the typical range for the selected material, so outliers are obvious without a separate chart.

Combining geometry, trig, and stockpile volume in one result panel removes the most common bookkeeping mistakes in introductory granular-flow problems.

Once the cone volume is in hand, the Gravel Calculator takes the next step by converting that volume and density into cubic yards and tons for a stockpile order.

The same material can pile at different angles depending on how it is poured and what condition it is in. These factors explain the gap between a textbook value and a measured pile.

• • The formula assumes a right circular conical pile. Real piles can be asymmetric or flattened by the floor surface, especially for cohesive materials.
• • The cone volume assumes a perfect right cone. Stockpiles poured against a wall flatten and hold less volume than a free cone.

These factors are not all equal. Grain shape sets the floor, moisture content sets the ceiling, and pouring method decides where between the two the actual angle lands.

According to the Engineering Toolbox - Angles of Repose, the same table lists dry sand at 35 degrees, damp sand at 40 degrees, gravel at 40 degrees, and broken and crushed stone near 30 degrees.

According to the Engineering Toolbox - Friction Coefficients, the static friction coefficient of a granular bed is usually larger than its kinetic value, which is why a freshly poured pile holds a steeper slope than a settled one.

The repose angle is one specific angle out of the broader trig toolbox, and the Trigonometry Calculator covers the related sine, cosine, and tangent identities used in pile geometry.