Tipping angle stability
An object with a 0.4 m base and its centre of gravity 0.8 m high tips over at 14.0362434679°. Enter the base width and the centre of gravity height to see how far it can lean before falling. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Base width
- 0.4
- Cg height
- 0.8
- Half base
- 0.2
- Tan theta
- 0.25
- Angle rad
- 0.2449786631
- Angle deg
- 14.0362434679
This tipping angle stability calculator finds the critical tilt at which a rigid object tips over about one edge of its base. You enter the full base width and the height of the centre of gravity above the base; the engine assumes the CG sits above the geometric centre and returns the angle where the vertical through the CG first leaves the supporting base. The closed form is θ equals arctan of half the base width divided by the CG height, with tan(θ) reported for checks against free-body sketches. A zero CG height is rejected because that geometry is undefined and would divide by zero. The path is pure deterministic arithmetic: no network, no sampling, and no date-dependent shortcuts.
What the tipping angle means and what you get
In elementary rigid-body statics, an object resting on a flat base remains stable while the vertical line through its centre of gravity intersects the base support polygon. When the body is tilted about one base edge, that vertical migrates toward the pivot edge. At the critical tip angle the vertical passes exactly through the edge; any further tilt places the line of action of weight outside the base, so the restoring moment reverses and the object falls over.
Wider bases and lower CG heights increase the tip angle and therefore improve static tip resistance; a tall narrow object tips at a small angle. Zero base width is accepted and yields a zero tip angle: a knife-edge base offers no restoring moment under tilt.
Where tip-angle checks fit in study, design, and software
Students meet tipping stability when a textbook contrasts a low sports car with a tall van, when a lab asks for the tilt table angle at which a block first rocks onto an edge, or when a contest problem gives base width and CG height and asks for the critical angle without calculus. Teachers can pre-generate answer keys for modest widths and heights without trusting opaque calculator apps that silently switch units or confuse half-width with full width.
Developers call the same contract from scripts to seed deterministic test vectors for simple stability demos, to normalize user-entered packaging geometry before a free-body diagram, or to satisfy programming-challenge statements that demand the tip angle rather than a hand-waved sketch. First-order design checks use θ equals arctan of half width over CG height when only base footprint and estimated mass centre height are known before full multi-body contact or seismic load models are built.
What you can do with it
Homework and lab tilt tables
Compute the critical tip angle from base width and CG height before grading worksheets or comparing to measured tilt-table results.
Packaging and first-order design checks
Estimate how far a tall box or appliance can lean before tipping from footprint width and estimated centre-of-mass height.
FAQ
What does the tipping angle stability calculator cost?
It is free in your browser on this page.
Does this model friction, acceleration, or multi-edge bases?
No. It models ideal quasi-static tipping of a rigid body about one base edge with the CG above the geometric centre. Dynamic loads, friction cones, and arbitrary support polygons are outside scope; use a full statics or multi-body model when those matter.
How is the tipping angle calculated?
Divide half the base width by the height of the centre of gravity and take the arctangent. A base of 0.4 m with the centre of gravity at 0.8 m tips at 14.0362434679°.
Why do a wider base and a lower centre of gravity help?
They raise the angle at which the object tips. The vertical line through the centre of gravity needs more tilt to pass the base edge, while a tall narrow object tips at a small angle.
What happens if the centre of gravity height is zero?
It is rejected. With a zero height the geometry is undefined and the calculation would divide by zero, so enter the real height of the centre of gravity above the base.
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/tipping-stability \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base_width":0.4,"cg_height":0.8}'const res = await fetch("https://api.kit.forhosting.com/mech/tipping-stability", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base_width": 0.4,
"cg_height": 0.8
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/tipping-stability",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base_width": 0.4,
"cg_height": 0.8
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/tipping-stability", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base_width":0.4,"cg_height":0.8}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base_width":0.4,"cg_height":0.8}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/tipping-stability", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"base_width": 0.4,
"cg_height": 0.8
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.tipping_stability",
"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
What tipping stability means and what the response contains
For a rectangular or otherwise symmetric base of full width base_width with the CG above the midpoint at height cg_height, elementary geometry gives tan of the tip angle equal to half_base over cg_height, so θ equals arctan of that ratio. This calculator returns the normalized inputs base_width and cg_height, the half_base used in the formula, tan_theta, the tip angle in both radians and degrees, the precision used for rounding, a method tag tipping_angle_from_base_and_cg, a short formula reminder, and a unit note that the angle is dimensionless in length units because the ratio cancels them. There is no multi-body contact model, no friction cone, and no numerical integration of rigid-body dynamics: the engine either evaluates the closed form inside the supported numeric range or rejects the input with invalid_input. Rejected cases include a missing width or height, non-finite values, magnitudes above the documented ceilings, a negative base width, and any CG height that is zero or non-positive, because the tip formula divides by the CG height.
How width and height are read and how the angle stays reproducible
Base width must be finite and non-negative. Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces and non-numeric tokens are rejected. The preferred field for the full base span is base_width, with aliases width, w, base, and B so spreadsheets, query strings, and typed forms can reuse the same contract. The preferred field for the centre-of-gravity height is cg_height, with aliases height, h, cog_height, center_of_gravity_height, centre_of_gravity_height, and H. The CG height must be finite and strictly positive: a zero or near-zero value fails with invalid_input and a message that names the field and explains that the tipping angle is undefined when the height is zero, so a bad cell does not become a silent nonsense result. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric outputs, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the tip-over formula yourself.
Where tip-angle checks fit in study, design, and software
Product and education tools use the endpoint when a learner types width and height and the UI must show degrees, radians, and tan of theta for class discussion. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with inclined-plane or friction helpers when you need related statics, and with other math endpoints when you are building a small lab of free, deterministic physics calculators.
Integration use cases
Deterministic test fixtures
Generate reproducible angle_deg, angle_rad, and tan_theta values for unit tests, stability demos, and CI assertions on tip-over geometry.
Technical questions
What formula is used for the tipping angle?
For a CG above the centre of a base of full width base_width and height cg_height: tan(θ) = (base_width/2)/cg_height, so θ = arctan((base_width/2)/cg_height). The response reports tan_theta, angle_rad, and angle_deg.
What happens if the centre-of-gravity height is zero?
The input is rejected with invalid_input. A zero CG height makes the tip geometry undefined and would divide by zero in tan(θ) = (base_width/2)/cg_height, so the task fails instead of returning a nonsense result.
Does a wider base or a lower CG improve tip resistance?
Yes. Increasing base_width or decreasing cg_height increases the critical tip angle θ, so a larger tilt is required before the vertical through the CG leaves the base.