Ground speed calculator
At 250 knots true airspeed with a 40-knot tailwind, ground speed is 290 knots; turn that wind into a headwind and it drops to 210. Enter the airspeed, the wind speed and the angle between them to plan time en route. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Airspeed
- 250
- Wind speed
- 40
- Angle deg
- 0
- Ground speed
- 290
- Tailwind component
- 40
Ground speed is how fast an aircraft actually moves over the Earth's surface. It is not the same as true airspeed, which is how fast the aeroplane moves through the surrounding air mass. Wind then adds a vector: a tailwind increases ground speed, a headwind reduces it, and a crosswind changes the resulting magnitude even when the heading is left uncorrected. This calculator takes three numbers — true airspeed, wind speed, and the included angle between the flight-path vector and the wind vector — and returns the ground-speed magnitude from the law of cosines in the wind triangle. Zero degrees is a pure tailwind, one hundred eighty degrees is a pure headwind, and ninety degrees is a pure crosswind on an uncorrected heading.
What ground speed is and how the wind triangle gives it
When an aircraft flies, two velocity vectors sit in the same horizontal plane. The true-airspeed vector points the way the nose is going through the air. The wind vector points the way the air mass itself is moving over the ground.
Their vector sum is the ground-velocity vector, and the length of that sum is ground speed — the number a GPS ground-speed readout reports, and the number you need for time-en-route. Expanding the squared length with the law of cosines yields a closed formula that never needs a sketch once you know the included angle θ between those two vectors: ground speed equals the square root of true airspeed squared plus wind speed squared plus twice true airspeed times wind speed times the cosine of θ.
When θ is zero the cosine is one, the two vectors are parallel, and ground speed collapses to true airspeed plus wind speed — a pure tailwind. When θ is one hundred eighty degrees the cosine is minus one, the vectors oppose, and ground speed is the absolute difference of the two speeds — a pure headwind.
When θ is ninety degrees the cosine vanishes and ground speed is the hypotenuse of true airspeed and wind speed, which is the uncorrected-heading crosswind case, not the crabbed case that holds a chosen track. The identity is unit-agnostic: knots, kilometres per hour, miles per hour, or metres per second all work as long as airspeed and wind speed share one unit. The returned ground speed uses that same unit.
This landing answers the search “ground speed calculator” with those three inputs. A heading-correction page that assumes a pure ninety-degree crosswind, or a track-from-heading page that wants meteorological wind-from and a true heading, is a different query and a different form.
When ground speed matters, and related tools
Flight students, dispatchers, and physics teachers all meet the same triangle with different vocabulary. A student checking a textbook problem that names true airspeed, a wind speed, and “the angle between the flight path and the wind” wants this page, not a full E6B solve that also asks for magnetic variation and a desired course. A dispatcher who already reduced METAR wind to a relative angle off the nose can paste three numbers and read ground speed without reconstructing headings.
Nearby pages must not cannibalize this query. The crosswind heading calculator assumes a pure ninety-degree wind and returns the crab angle plus the track-holding ground speed equal to the square root of true airspeed squared minus wind squared, which is a different identity. The true-ground-track-from-heading calculator wants a heading, a meteorological wind-from direction, true airspeed, and wind speed, and it returns both track and ground speed from north and east components.
Top-of-descent tools take ground speed as an input; they do not compute it. Use those pages when that is what you searched for; use this one when you already have the included angle and want only ground speed. There is no network lookup, no clock, and no random draw.
The calculator does not convert units, apply density altitude, or solve for the heading that would hold a chosen track. Those are separate jobs.
What you can do with it
Classroom wind-triangle problems
Students enter true airspeed, wind speed, and the angle between flight path and wind to check the law-of-cosines ground speed against a textbook worked example.
Quick en-route ground-speed estimate
A planner who already has TAS and a relative wind angle off the nose reads ground speed in the same unit without reconstructing headings or a full E6B triangle.
FAQ
How do I calculate aircraft ground speed from true airspeed and wind?
Send non-negative true airspeed, non-negative wind speed, and the included angle in degrees between the flight-path vector and the wind vector. Ground speed is sqrt(TAS² + W² + 2·TAS·W·cos θ). Zero degrees is a tailwind; 180 degrees is a headwind.
What does the angle between flight path and wind mean?
It is the included angle between the true-airspeed vector (the way the nose is going through the air) and the wind vector (the way the air mass is moving). Zero means the wind blows with the nose; 180 means it blows against the nose; 90 is a pure crosswind on an uncorrected heading.
What happens if airspeed or wind speed is negative?
Direction is encoded by the angle, not by a signed speed, so a negative airspeed or wind speed is rejected and is not billed.
Does this calculator crab the heading to hold a track?
No. It adds the TAS and wind vectors at the angle you supply and returns the magnitude. The crabbed, track-holding identity (WCA = asin(W/TAS) and GS = sqrt(TAS² − W²) for a pure crosswind) is a different page.
Which speed units can I use?
Any unit, such as knots, kilometres per hour, miles per hour or metres per second, as long as airspeed and wind speed use the same one. The ground speed comes back in that same unit.
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/ground-speed \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"airspeed":250,"wind_speed":40,"angle_deg":0}'const res = await fetch("https://api.kit.forhosting.com/phys/ground-speed", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"airspeed": 250,
"wind_speed": 40,
"angle_deg": 0
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/ground-speed",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"airspeed": 250,
"wind_speed": 40,
"angle_deg": 0
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/ground-speed", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"airspeed":250,"wind_speed":40,"angle_deg":0}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"airspeed":250,"wind_speed":40,"angle_deg":0}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/ground-speed", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"airspeed": 250,
"wind_speed": 40,
"angle_deg": 0
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.ground_speed",
"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
Negative airspeed or negative wind speed is rejected as invalid_input rather than a nonsense speed. Successful calls cost the published $0.002; failed validation is not billed.
Input contract, angle convention, and error behaviour
Zero degrees means the wind blows with the nose (pure tailwind). One hundred eighty degrees means it blows against the nose (pure headwind). Ninety degrees is a pure crosswind on an uncorrected heading. Angles outside the interval from zero to three hundred sixty are accepted because cosine is 360-degree periodic, so minus thirty and three hundred thirty produce the same ground speed. Zero airspeed is allowed and means the aircraft is not flying through the air, so ground speed equals the wind-vector magnitude. Zero wind is allowed and ground speed equals true airspeed. The same speeds at 180 degrees yield 210. The same speeds at 90 degrees yield the hypotenuse of 250 and 40. POST a JSON object with three required finite numbers. The field airspeed is true airspeed; aliases tas, true_airspeed, air_speed, and va are accepted. The field wind_speed is the wind-vector magnitude in the same unit; aliases wind, ws, and wind_vel are accepted. The field angle_deg is the included angle in degrees between the flight-path vector and the wind vector; aliases angle, theta, and wind_angle are accepted. Airspeed and wind speed must not be negative: a negative speed is not a direction — the angle already encodes direction — and the request fails with invalid_input naming the field. Missing fields, empty strings, non-numeric text, NaN, and Infinity fail with invalid_input and are not billed. On success the body echoes the three inputs after stable twelve-decimal rounding, reports ground_speed as the primary result, reports tailwind_component as wind_speed times cosine of the angle (positive for a tailwind, negative for a headwind), and restates the formula so a client can print it next to the number. Example: airspeed 250, wind_speed 40, and angle_deg 0 yield ground speed 290 in the same unit. Pricing for a successful item is a flat $0.002.
Where a ground-speed API earns its keep, and nearby tools
Scripted lesson generators need a deterministic oracle: identical inputs must produce identical JSON so an answer key cannot drift after a library upgrade.
Integration use cases
Deterministic fixtures for aviation helpers
Pipelines freeze JSON goldens for tailwind, headwind, and crosswind angles so a refactor cannot change the vector-addition contract silently.
Technical questions
What does the ground speed calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.