About the Angle of Repose
The angle of repose is the steepest angle, measured from the horizontal, at which a heap of loose granular material remains stable without slumping further. Pour dry sand, gravel, sugar, or grain onto a flat surface and it naturally settles into a cone whose sides sit at a characteristic slope — that slope angle is the angle of repose. It is a basic property used across geotechnical engineering, mining, agriculture, and bulk materials handling.
Understanding the formula
For an ideal cone-shaped pile, the height h, base radius r, and angle of repose θ form a right triangle from the apex straight down to the base center, then out to the base edge. That gives tan(θ) = h / r, so θ = arctan(h / r). Because base diameter d is usually easier to measure than radius, and r = d/2, this calculator uses the equivalent form θ = arctan(2h / d). The tangent of that same angle, tan(θ), is also the effective coefficient of friction between the grains — a steeper pile means more inter-particle friction and interlocking.
Measuring height and diameter
- Let the pile come completely to rest — grains still rolling down the slope will bias the measurement.
- Measure height (h) vertically from the base to the apex, not along the slanted surface.
- Measure base diameter (d) straight across the widest point of the footprint; average two or three readings if the base is not a perfect circle.
- Height and diameter must use the same length unit — the ratio 2h/d is unitless, so meters, feet, or inches all work as long as both inputs match.
Knowing the limits
This model assumes an ideal, symmetric cone of dry, free-flowing, cohesionless grains poured under gravity alone. It does not hold well for sticky or wet cohesive materials (which can stand at unrealistically steep or irregular slopes), for piles disturbed by vibration or wind, or for materials that segregate by particle size as they fall (larger grains rolling further, changing the cone's shape). The angle measured also depends slightly on how the pile was formed — a poured pile and a pile left behind after material drains away from it can differ by a few degrees.