Sinusoidal projection calculator
Place a point on the equal-area sinusoidal map: 45° north and 30° east on a unit sphere become x 0.3702402448 and y 0.7853981634. Each parallel is scaled by the cosine of its own latitude, so areas stay true. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Latitude
- 45
- Longitude
- 30
- Radius
- 1
- Central meridian
- 0
- X
- 0.3702402448
- Y
- 0.7853981634
The sinusoidal projection, also called the Sanson–Flamsteed projection, is a classic equal-area map projection that places each parallel of latitude as a straight horizontal line and scales the length of that parallel by the cosine of latitude so that areas on the sphere are preserved on the plane. This sinusoidal projection calculator accepts a geographic latitude and longitude in decimal degrees, optional sphere radius and central meridian, and returns plane coordinates x and y together with intermediate radians and cos(latitude) so you can verify the equal-area construction by hand.
What the sinusoidal equal-area projection computes
On a sphere of radius R the infinitesimal area element is R squared times cos(φ) dφ dλ, where φ is latitude and λ is longitude in radians. The sinusoidal projection maps that element onto the plane with x equals R times (λ minus λ0) times cos(φ) and y equals R times φ, so the Jacobian of the map matches the spherical area factor and regions keep true relative area.
Meridians become sinusoidal curves that meet at the poles, while parallels remain equally spaced straight lines; the central meridian is a straight vertical axis when λ0 is chosen as the reference longitude. Unlike Web Mercator, which preserves angles and inflates high-latitude area, or cylindrical equal-area variants that compress north–south spacing with a fixed cosine at a standard parallel, the pure sinusoidal form scales every parallel by its own cos(φ).
This calculator implements that forward project only: you supply latitude in the closed interval from negative ninety to positive ninety degrees and longitude from negative one hundred eighty to positive one hundred eighty, plus optional R greater than zero and central meridian λ0.
What you can do with it
Cartography homework and labs
Project textbook lat/lon pairs into sinusoidal x,y with cos(φ) scaling for equal-area map exercises and answer keys.
FAQ
Is the sinusoidal projection equal-area?
Yes. The map Jacobian matches the spherical area element R² cos(φ) dφ dλ, so regions keep correct relative area (shapes distort especially far from the central meridian).
What latitude and longitude ranges are accepted?
Latitude must be in [-90, 90] degrees and longitude in [-180, 180].
What does the central meridian change?
Longitudes are measured from it before the cosine scaling, so it becomes the straight vertical axis of the map. It defaults to 0°, the Greenwich meridian, and accepts any value from -180 to 180.
Which radius should I enter?
Leave the default of 1 for unit-sphere map units, or enter any positive radius, such as a mean Earth radius in kilometres or metres, to get x and y in those units.
How is this different from Web Mercator?
Web Mercator keeps angles but inflates areas at high latitudes. The sinusoidal projection scales every parallel by the cosine of its own latitude instead, so countries and continents keep their true relative area.
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/geo/sinusoidal-project \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"latitude":45,"longitude":30}'const res = await fetch("https://api.kit.forhosting.com/geo/sinusoidal-project", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"latitude": 45,
"longitude": 30
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geo/sinusoidal-project",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"latitude": 45,
"longitude": 30
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geo/sinusoidal-project", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"latitude":45,"longitude":30}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"latitude":45,"longitude":30}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geo/sinusoidal-project", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"latitude": 45,
"longitude": 30
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geo.sinusoidal_project",
"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 the sinusoidal equal-area projection computes
The response echoes the inputs, reports x and y, lat_rad and lon_rad, cos_lat, precision, a short formula string, and the method tag sinusoidal_project so clients can log which equal-area model produced the point without re-deriving the trig by hand.
Inputs, ranges, rounding, and error behaviour
Preferred field names are latitude and longitude in decimal degrees; aliases lat and phi for latitude and lon, lng, long, and lambda for longitude are accepted so spreadsheets and typed forms can reuse the same contract. Optional radius (alias R) defaults to one for unit-sphere map units; any positive finite value is allowed when you work in kilometres or metres of a mean Earth radius. Optional central_meridian (aliases lon0 and lambda0) defaults to zero degrees so longitudes are measured from Greenwich before the cos(φ) scale is applied. Values outside latitude plus or minus ninety, longitude plus or minus one hundred eighty, non-positive radius, non-finite numbers, or missing required angles fail with invalid_input and are not billed.
Where a sinusoidal projection API earns its keep
Cartography students meet the sinusoidal projection when a lab asks for equal-area plane coordinates without the standard-parallel bookkeeping of Gall–Peters or Albers, and teachers need reproducible x and y for modest city coordinates instead of opaque desktop GIS clicks. Data journalists and educators who compare country or continent areas side by side often prefer a true equal-area base; generating projected points through a tiny parse API avoids pulling a full PROJ stack for a single forward formula. 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. Pair it with Web Mercator or Gall–Peters tools when you need other map models, and keep this endpoint for the pure sinusoidal equal-area contract that scales longitude by cos(latitude).
Integration use cases
Equal-area GIS fixtures
Generate deterministic plane coordinates for unit tests that assert area-preserving transforms without a full PROJ install.
Teaching demos and widgets
Show how meridians curve and parallels shrink with latitude using the free browser path and the same API contract.
Technical questions
What formulas define the sinusoidal projection?
With sphere radius R and central meridian λ0, x = R · (lon_rad − lon0_rad) · cos(lat_rad) and y = R · lat_rad. That cos(latitude) factor preserves area.
What is the default radius and central meridian?
Radius defaults to 1 (unit sphere map units) and central_meridian defaults to 0° (Greenwich). Pass R or radius and central_meridian when you need other units or a shifted meridian.
What does the sinusoidal projection calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.