Bank angle calculator
At 120 m/s on a 1,500 m radius, a level turn needs a bank of just under 45°. Enter airspeed and radius in coherent units and read the angle in degrees and radians, plus the load factor. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Speed
- 120
- Radius
- 1,500
- G
- 9.81
- Tan theta
- 0.978593
- Bank angle rad
- 0.774579
- Bank angle deg
- 44.380132
You send the true airspeed and the turn radius, and it returns the required bank angle in degrees, the same angle in radians, the tangent of the angle, and the resulting load factor on the airframe.
Why bank angle is fixed by speed and radius
In a level turn the lift vector does two jobs at once: its vertical component must equal the weight so the aircraft neither climbs nor descends, and its horizontal component is the centripetal force that curves the flight path. Dividing those two requirements eliminates the lift itself and leaves a relation between just three quantities — speed, turn radius, and gravity.
The result, tan(theta) = v^2 / (r g), means the pilot does not get to choose the bank angle independently: once the airspeed and the desired radius are set, there is exactly one bank angle that keeps the turn level and coordinated. Fly slower or accept a wider radius and the required bank shrinks; fly faster or tighten the radius and it grows steeply, because the tangent rises with the square of the speed.
This is why procedure turns, holding patterns, and circling approaches are all specified with strict speed limits — the speed is really a limit on the bank angle needed to stay inside the protected airspace.
How to use the calculator and read the output
Send two required fields — speed and radius — and optionally the gravitational acceleration g, which defaults to 9.81 m/s^2. Any coherent unit system works: metres per second with metres, knots with nautical miles, or kilometres per hour with kilometres, as long as you use the same length and time units throughout and g matches them. Every result is rounded to six decimals so automated comparisons stay stable across machines.
Where the bank angle calculation fits in practice
Flight-planning tools use this relation to check whether a procedure turn at the maximum permitted speed stays within the bank limits of the aircraft category. Simulation and game developers use it to bank aircraft models realistically instead of animating an arbitrary roll.
UAV mission planners invert it to choose a loiter speed that keeps the turn radius inside a survey area, and instructors use it to show students how quickly the required bank — and with it the load factor and the stall speed — climbs when a steep turn is flown too fast.
What you can do with it
Check a procedure turn
Verify that the required bank at the maximum procedure speed stays within the aircraft's limits before flying it.
Plan a UAV loiter
Pick a speed whose turn radius fits the survey area, and read off the bank angle the autopilot must hold.
Teach level-turn physics
Show students how required bank and load factor grow with the square of speed for a fixed radius.
FAQ
Which units should I use?
The angle itself is unit-independent as long as speed, radius and g are coherent.
What counts as invalid input?
A speed or radius that is missing, zero, negative, or not a finite number.
Does it assume a coordinated turn?
Yes. It solves the classical level, coordinated-turn relation tan(theta) = v^2 / (r g). It does not model sideslip, wind, or climb.
Why does the required bank grow so quickly with speed?
Because the tangent of the bank angle rises with the square of the speed. Fly faster or tighten the radius and the bank steepens sharply; slow down or accept a wider turn and it shrinks.
What does the load factor in the result mean?
It is one divided by the cosine of the bank angle, and it tells you how many g the wings must support in the turn. As the bank steepens, the load factor and the stall speed climb with it.
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/phys/bank-angle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"speed":120,"radius":1500}'const res = await fetch("https://api.kit.forhosting.com/phys/bank-angle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"speed": 120,
"radius": 1500
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/bank-angle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"speed": 120,
"radius": 1500
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/bank-angle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"speed":120,"radius":1500}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"speed":120,"radius":1500}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/bank-angle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"speed": 120,
"radius": 1500
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.bank_angle",
"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
The bank angle API answers the first question of any turn: how steeply must the aircraft bank to hold a level, coordinated turn of a given radius at a given speed? The physics is the classical level-turn relation tan(theta) = v^2 / (r g), evaluated exactly and deterministically, so the same inputs always produce the same answer — in the paid API and in the free browser version on this page.
How to use the calculator and read the output
The response echoes the validated inputs and returns tan(theta), the bank angle in radians, the bank angle in degrees — the number a pilot or autopilot actually cares about — and the load factor n = 1 / cos(theta), which tells you how many g the wings must support in the turn. Both speed and radius must be strictly positive numbers; zero, negative, missing, or non-numeric values are rejected with an invalid_input error and are never charged.
Where the bank angle calculation fits in practice
Because the endpoint is a pure deterministic computation with no network calls, no stored state, and no randomness, it is safe to call inside larger pipelines: the same code runs free in your browser on this page, and you pay $0.002 per request only when you automate it through the API.
Technical questions
What does it cost?
$0.002 per request. It is also free to run in your browser on this page — the same code computes both.
What is the load factor in the response?
It is n = 1 / cos(theta), the number of g the wings support in the turn — 1.41 at 45 degrees of bank, 2.0 at 60 degrees.
Can I change the gravity value?
Yes. Pass g (alias gravity) in units coherent with your speed and radius. It defaults to 9.81 m/s^2 and must be positive.