Angle of repose calculator
A static friction coefficient of 0.5 means a block starts to slide at 26.565051177078 degrees: the calculator takes the arctangent of μₛ and gives the angle of repose in degrees and radians, with every step shown. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mu s
- 0.5
- Mu
- 0.5
- Static friction
- 0.5
- Coefficient
- 0.5
- Tan theta
- 0.5
An angle of repose calculator turns the elementary limiting-equilibrium identity into a practical tool for homework, lab books, and quick engineering checks. Enter the dimensionless static friction coefficient μₛ for a block on an incline or for a granular pile surface pair, and the tool returns the maximum slope angle θ before sliding begins using θ equals arctan of μₛ, together with the same angle in radians and degrees, a substituted equation, a short physics message, and ordered step strings you can paste into worksheets or automated tests. Missing the coefficient, non-finite values, and out-of-range magnitudes are likewise rejected before any arctangent is published.
What θ = arctan(μₛ) means and when the coefficient must not be negative
The angle of repose is the steepest incline angle at which an object remains at rest under static friction, or equivalently the natural slope angle of a cohesionless granular pile. For a rigid block on a rough plane, free-body balance at the verge of slip equates the downhill component of weight to the limiting friction force: m g sin(θ) equals μₛ times m g cos(θ). Canceling mass and gravity yields tan(θ) equals μₛ, so θ equals arctan of μₛ.
The same algebraic relation appears when a granular heap is built until the free surface fails and grains begin to cascade. The result is an angle: this calculator reports both radians from the principal Math.atan branch and degrees via the usual one-hundred-eighty-over-pi conversion, so classroom answers that expect degrees and engineering codes that prefer radians both get a first-class field. Zero is valid and maps to a frictionless horizontal limit of zero degrees.
Very large positive coefficients drive the angle toward ninety degrees asymptotically, which is the mathematical limit of arctan at positive infinity and matches the intuition that extreme stickiness supports nearly vertical faces until other failure modes dominate. This landing does not invent material tables, moisture cohesion, or velocity-dependent kinetic friction: you supply the operative static coefficient for your trial, and the solver returns only the corresponding geometric angle.
What you can do with it
Homework checks for θ = arctan(μₛ)
Students enter a static friction coefficient from textbook incline problems and compare the degree and radian angles plus substitution steps against a deterministic result before submitting.
Lab pre-check before material tables
Pipelines convert a measured or assumed static coefficient into the critical slope angle first, then compare that angle against observed pile or ramp failure only after the arctangent check matches the lab notes.
FAQ
What formula does the angle of repose calculator use?
It uses θ = arctan(μₛ), where μₛ is the static friction coefficient. The angle is returned in both radians and degrees.
What happens if the static friction coefficient is negative?
A negative μₛ is rejected because the angle of repose is undefined for non-physical Coulomb coefficients below zero.
Is μₛ = 0 allowed?
Yes. Zero static friction yields an angle of repose of zero degrees (and zero radians), corresponding to a frictionless horizontal limit.
Is this the same as a granular pile angle of repose?
The same tan(θ) = μₛ relation is used for both a block on an incline at limiting equilibrium and a cohesionless granular free surface. Cohesion, moisture, and shape effects are outside this pure arctangent model.
Is the angle of repose calculator free?
The browser tool is free and runs locally.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/mech/angle-of-repose \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mu_s":0.5}'const res = await fetch("https://api.kit.forhosting.com/mech/angle-of-repose", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mu_s": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/angle-of-repose",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mu_s": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/angle-of-repose", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mu_s":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mu_s":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/angle-of-repose", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mu_s": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.angle_of_repose",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
If the coefficient is negative, validation fails with a clear invalid_input error so non-physical Coulomb models never silently invent a slope. Pricing is a flat $0.002 per successful API request; failed validation is never charged.
What θ = arctan(μₛ) means and when the coefficient must not be negative
A negative static friction coefficient has no meaning in classical Coulomb contact models—friction cannot reverse the sign of the normal reaction into a driving force—so this tool rejects μₛ less than zero with invalid_input rather than returning a negative slope or a quiet NaN.
How to call the API and what comes back
Absolute magnitude must stay within the published limit of one trillion, and the value must not be negative. Send a JSON body with the required finite field mu_s, which also accepts the aliases mu, coefficient, static_friction, mu_static, and us for form paste paths and multi-language homework sheets. Failed validation—missing coefficient, non-finite numbers, values outside absolute limits, or a negative μₛ—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer JSON numbers when you control the client; string numbers are accepted only when they match a strict finite numeric pattern for homework-form paste paths. Round-trip checks are easy: recompute arctan of your coefficient by hand or in a spreadsheet and confirm it equals the published angle_rad within the same twelve-decimal rounding used in the response.
Practical tips, limits, and related searches
Use a dimensionless coefficient end to end—this landing does not convert force pairs into μ, does not convert degrees to radians for you beyond the published dual fields, and does not look up rubber-on-concrete or sand-on-steel handbook tables. When teaching Coulomb friction on an incline, treat this tool as the geometry step only: if the source problem gives friction force and normal force instead, first form μ equals Ff over N with a companion calculator, then pass that coefficient here. Prefer this capability when you already know the static coefficient and only need the critical incline angle; if your lab measures limiting angle directly and wants μ back, invert the relation as tan of the measured angle rather than calling this solver in reverse. Keep μₛ inside the published absolute range so the arctangent remains finite in IEEE double precision on the Worker and in the browser. Pair classroom examples deliberately: μₛ equals zero point five yields about twenty-six point five seven degrees; μₛ equals one yields exactly forty-five degrees; and a negative coefficient demonstrates the hard invalid_input path required by the product brief. This page targets people searching for an angle of repose calculator, maximum slope angle from static friction, arctan of mu_s, or granular pile slope online; it is not a multi-body DEM simulator, not a cohesive soil strength chart, and not a kinetic friction time-of-slide integrator. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately set a negative coefficient to exercise the error contract.
Integration use cases
Unit tests for game and sim incline helpers
Generate known-good μₛ and angle pairs and deliberate negative-coefficient rejections for coding kata expected values in physics engines and sliding contact stubs.