Meridian radius of curvature
Get the radius of curvature of the meridian at any latitude on WGS84: at 45° it is about 6,367,382 m, so one degree of latitude covers about 111.13 km there. Handy for turning latitude steps into ground metres. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Worked example
- Latitude
- 45
- Meridional radius m
- 6,367,381.815619548
- Meridional radius km
- 6,367.3818156195
- Meters per degree
- 111,131.7774141756
- Ellipsoid
- WGS84
The meridional radius of curvature is the radius of the ellipse that best fits the Earth meridian at a chosen geodetic latitude. On the WGS84 ellipsoid it is written M of phi equals a times one minus e squared, all over the three-halves power of one minus e squared times sin squared of phi. Surveyors, GIS engineers, and navigation software use M to convert small latitude differences into ground metres along a meridian, because north–south distance scales as M times d-phi in radians. This meridian radius of curvature calculator accepts a latitude in decimal degrees, applies the closed-form WGS84 constants, and returns M in metres and kilometres plus the derived metres-per-degree figure for one degree of latitude.
What the meridional radius of curvature measures on WGS84
An oblate ellipsoid of revolution is defined by a semi-major axis a in the equatorial plane and a flattening f that shortens the polar semi-minor axis. WGS84 fixes a at six million three hundred seventy-eight thousand one hundred thirty-seven metres and the reciprocal of f at two hundred ninety-eight point two five seven two two three five six three. From those constants the first eccentricity squared is e squared equals f times two minus f.
At a geodetic latitude phi the meridional radius of curvature M is the radius of curvature of the meridian ellipse in the north–south plane. Algebraically M equals a times one minus e squared, divided by the quantity one minus e squared sin squared phi raised to the three-halves power. At the equator sin phi is zero so M is smallest; at the poles sin phi is plus or minus one so M is largest.
That variation is exactly why a degree of latitude is slightly longer near the poles than near the equator.
Where the meridian radius of curvature helps
Mapping and GIS pipelines convert pixel or grid steps into ground metres and need the local north–south scale factor at the working latitude without pulling a full geodesy suite. Survey education and exam generators need a deterministic oracle for M of phi so students can check hand calculations against WGS84. Invalid latitude is rejected before any arithmetic and does not consume balance.
Keep requests small, pin the rounded metres in tests, and you get reproducible ellipsoid radii for notebooks, LMS plugins, and edge workers without standing up a symbolic or network geodesy stack of your own.
What you can do with it
Survey and geodesy homework checks
Let students submit a geodetic latitude and verify the closed-form meridional radius against textbook WGS84 constants and intermediate eccentricity terms.
Navigation short-leg north–south estimates
Approximate short meridian distances as M times delta-phi in radians inside flight, marine, or field apps without embedding a full geodesic library.
FAQ
What is the meridian radius of curvature on WGS84?
It is M(φ)=a(1−e²)/(1−e²sin²φ)^{3/2} with WGS84 a and f.
How does M relate to the length of a degree of latitude?
The ground length of one degree of latitude is M(φ)·π/180.
Is this the same as geocentric radius or prime vertical N?
No. Geocentric radius is the centre-to-surface distance; prime vertical N governs east–west scaling. M is the north–south meridional radius of curvature only.
Why is the meridian radius smallest at the equator and largest at the poles?
Because the Earth is a flattened ellipsoid. In the formula the squared sine of the latitude is zero at the equator and one at the poles, so the radius grows toward the poles and a degree of latitude gets slightly longer.
Which ellipsoid constants does the calculator use?
The WGS84 values: a semi-major axis of 6,378,137 m and an inverse flattening of 298.257223563. The eccentricity squared is derived from the flattening, and all constants are shown with the result so you can check them.
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/radius-meridian-curvature \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"latitude":45}'const res = await fetch("https://api.kit.forhosting.com/geo/radius-meridian-curvature", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"latitude": 45
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geo/radius-meridian-curvature",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"latitude": 45
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geo/radius-meridian-curvature", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"latitude":45}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"latitude":45}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geo/radius-meridian-curvature", 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
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geo.radius_meridian_curvature",
"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
Latitude outside the closed interval from minus ninety to ninety fails with invalid_input.
What the meridional radius of curvature measures on WGS84
The capability returns M itself in metres and kilometres, echoes the WGS84 constants used, and also reports metres_per_degree equal to M times pi over one hundred eighty so clients that think in degree increments need no extra conversion. Optional aliases lat and phi map to the same required latitude field.
Input contract, response fields, and validation errors
Example: latitude forty-five degrees yields a meridional radius near six million three hundred sixty-seven thousand three hundred eighty-two metres and about one hundred eleven point one three kilometres per degree of latitude. POST a JSON object with one required finite number: latitude, the geodetic latitude in decimal degrees on WGS84, allowed on the closed interval from minus ninety through ninety. Numeric strings that parse as finite numbers are accepted; empty strings, missing keys, non-finite values, and latitudes outside the interval fail with invalid_input naming the problem, and those failures are not billed. On success the body includes the echoed latitude; meridional_radius_m and meridional_radius_km as stably rounded floats; meters_per_degree for north–south ground length of one degree at that latitude; ellipsoid set to the string WGS84; semi_major_m, flattening, f_inv, and eccentricity_squared so you can audit the constants without looking them up; and a formula string restating the closed form used. Pair the radius with your own arc-length integrator if you need multi-degree meridian distances; this endpoint evaluates M at a single phi only.
Where a meridian radius of curvature API earns its keep
Navigation and flight-planning tools that approximate short legs along a meridian use M times delta-phi as a first-order distance; calling a tiny parse API avoids embedding ellipsoid constants in every microservice.
Integration use cases
GIS and map scale at a working latitude
Convert a small latitude step on a grid or raster into ground metres using M(φ)·dφ, or use meters_per_degree for one-degree tiles on WGS84.
Technical questions
Which input does the calculator need?
A single finite latitude in decimal degrees from −90 to 90 (aliases lat and phi). Outside that range or a non-finite value fails with invalid_input.
What does the meridian radius of curvature calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.