ForHosting KIT · Math

Distance to horizon calculator

With your eyes 1.7 m above the surface, the geometric horizon lies about 4.6541812266 km away, or 2.51305681782 nautical miles. Height, planet radius and units are adjustable, and the result is the straight line of sight, not the curved surface arc. Free in your browser.

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Run — free How it works & API

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Worked example

If you enter

Height: 1.7
Height unit: M
Distance unit: Km

You get

Height m: 1.7
Earth radius: 6,371,000
Earth radius unit: M
Earth radius m: 6,371,000

How far can you see over open water or flat ground before the Earth curves away? This distance to horizon calculator answers that classic geometry question with the exact tangent formula used in surveying, sailing, and introductory physics. Enter your eye height above the surface, optionally override the planetary radius, choose input and output length units, and receive the straight-line range to the geometric horizon.

What the geometric horizon is and why the tangent formula works

Imagine a perfect sphere of radius R and an observer whose eyes sit a height h above its surface. The farthest surface point still visible along a straight line of sight is the point where a ray from the eye is tangent to the sphere. Drawing the radius to that tangent point forms a right triangle: one leg is R, the hypotenuse is R plus h, and the other leg is the tangent distance d from the eye to the horizon.

Pythagoras then gives d squared equals (R plus h) squared minus R squared, which simplifies to two R h plus h squared. Taking the positive square root yields d equals the square root of two R h plus h squared. That length is the straight-line (chord through air) range, not the arc length along the surface; for typical human or tower heights the two numbers are close, but this calculator always returns the tangent range because that is the quantity the geometric construction produces.

Setting h to zero returns zero range. Raising h from eye level to a cliff top or mast increases the horizon roughly with the square root of height, which is why a few extra metres of elevation buy a surprising amount of extra sea horizon. The model assumes a smooth sphere with no hills, buildings, haze, or atmospheric refraction; real visibility can be shorter or, with super-refraction, longer than this pure geometric bound.

Who uses the distance to the horizon

Sailing and coastal education use a distance to horizon calculator so students can turn eye height or freeboard into a planning radius over open water before discussing haze and swell. Hiking and mountain-guide materials convert lookout elevation into an approximate maximum sea or plain horizon for map annotations and safety briefings. Photography and drone teams convert camera altitude into a geometric ground-range limit for framing discussions over flat terrain.

Content sites that rank for distance to horizon calculator, how far can I see from height, or geometric horizon range benefit from a live free browser tool whose math matches the automation path byte for byte. Keep requests small, pin the cleaned floats in tests, and you get reproducible geometric ranges for lessons, apps, and edge workers without standing up a full Earth-curvature or radio-propagation stack of your own.

What you can do with it

Coastal and sailing education

Convert bridge, mast, or eye height into geometric sea-horizon range for classroom demos and open-water planning notes.

Lookout and hiking map annotations

Turn cliff or tower elevation into an approximate maximum plain or sea horizon for guides and briefing materials.

FAQ

How is distance to the horizon calculated from observer height?

For height h on a sphere of radius R the geometric tangent range is d=sqrt(2·R·h+h²).

Is the result the surface arc or the straight-line range?

It is the straight-line tangent length from the eye to the tangent point. For ordinary heights that is very close to the surface arc, but the formula always returns the tangent range.

What height should I enter?

Enter eye height above the local surface (or mean sea level when treating the sea as a sphere). Standing adults are often about 1.5–1.8 m; use tower or cliff elevation for elevated lookouts.

Does this include atmospheric refraction?

Refraction can extend or reduce real optical range; for mutual radio horizons with a k-factor, use a dedicated mutual-visibility tool.

How far is the horizon with my eyes 1.7 m above the ground?

About 4.6541812266 km, or 2.89197413766 miles, over a smooth Earth. The range grows roughly with the square root of your height, so a little extra elevation adds a surprising amount of horizon.

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

POSThttps://api.kit.forhosting.com/trig/horizon-distance

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/trig/horizon-distance \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"height":1.7}'

const res = await fetch("https://api.kit.forhosting.com/trig/horizon-distance", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "height": 1.7
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/trig/horizon-distance",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "height": 1.7
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/horizon-distance", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"height":1.7}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"height":1.7}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/horizon-distance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "height": 1.7
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "trig.horizon_distance",
  "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

Per request$0.002

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

Failed validation is not billed; successful calls use prepaid KIT balance at the published base price.

Input contract, units, response shape, and error behaviour

Example: height one point seven metres with default Earth radius yields a horizon near four point six six kilometres. Height thirty metres yields about nineteen point six kilometres. POST a JSON body with one required non-negative finite number: height, the observer eye height above the local surface. Optional height_unit defaults to metres and accepts m, ft, km, mi, or nmi. Optional earth_radius defaults to six million three hundred seventy-one thousand metres, the common mean Earth radius, with optional earth_radius_unit using the same length vocabulary. Optional distance_unit defaults to kilometres for the primary horizon_distance field. On success the body echoes height and units, reports height_m and earth_radius_m for SI audit, returns horizon_distance in the chosen unit, and always includes horizon_distance_m, horizon_distance_km, horizon_distance_mi, and horizon_distance_nmi so multi-unit UIs need no second conversion pass, plus a formula string and a short assumptions note. Missing height, negative height, non-positive Earth radius, non-finite values, and unknown unit strings fail with invalid_input naming the problem and are not billed. The same pure path powers the free browser tool and the paid API so goldens and production never diverge on rounding.

Integration use cases

Homework and GIS regression tests

Pin d=sqrt(2Rh+h²) with multi-unit outputs so algebra, surveying, and notebook checks stay deterministic.

Technical questions

What does the distance to horizon calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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