Spherical triangle area from excess
Get a spherical triangle's area straight from its excess, so on a sphere of radius 1 an excess of 0.158407 radians covers an area of 0.158407. You also see the excess in degrees and the share of the whole sphere. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Excess
- 0.158407
- Radius
- 1
- Excess rad
- 0.158407
- Excess deg
- 9.076053
- Excess unit
- rad
- Area
- 0.158407
When you already know the spherical excess of a triangle on a sphere, Girard's theorem turns that single number and the sphere radius into the triangle's surface area with one multiplication: area equals R squared times E, with E in radians. Surveyors, GIS engineers, astronomy teachers, and students of spherical trigonometry often hold the excess from a prior angle sum (A plus B plus C minus pi) or from a published table and only need the area step—without re-entering the three interior angles. This calculator accepts the excess and the radius, optionally converts degrees to radians, rejects non-positive excess and non-positive radius, and returns the area in the squared length unit of R together with the excess in both radians and degrees and the fraction of the full sphere surface.
Girard's theorem: area equals R squared times spherical excess
On a unit sphere the area of a spherical triangle equals its spherical excess E, measured in steradians. On a sphere of any positive radius R the surface scales with R squared, so Girard's classical identity is area equals R squared times E. The excess itself is defined as the surplus of the interior angle sum over a half-turn: E equals A plus B plus C minus pi when the angles are in radians.
Plane triangles always sum to exactly pi and therefore carry zero excess and zero curvature-based area contribution; spherical triangles sum to more than pi, and that surplus is proportional to how much of the sphere they cover. Search intent for this tool clusters around phrases such as spherical triangle area from excess, Girard theorem calculator, and area equals R squared E—not the separate problem of recovering E from three angles (that is a different landing).
You pass the excess you already have and the radius of the sphere you care about: Earth radius in kilometres for survey work, unit sphere for textbook solid-angle problems, or any consistent length unit for custom globes.
Where an area-from-excess calculator earns its keep
Geodesy and cadastral workflows often compute excess from measured angles in a separate step or import it from a field book, then need only the area product for a chosen Earth radius. Classroom and exam platforms that teach Girard's theorem want a deterministic oracle that marks wrong a non-positive excess instead of silently returning zero area that a buggy grader might treat as success. Astronomy and navigation notes use the sphere-fraction field to show how large a celestial spherical triangle is relative to the full sky.
Failed validation does not consume balance. Keep requests small, pin rounded outputs in tests, and you get reproducible areas for homework, LMS plugins, survey scripts, and edge workers without standing up a full spherical-geometry library of your own.
What you can do with it
Survey and geodesy area from known excess
Convert a field-measured or precomputed spherical excess and a chosen Earth radius into triangle surface area in consistent R² units.
GIS spherical-triangle area sum
Sum Girard areas from per-triangle excess values when a region has already been partitioned into spherical triangles.
FAQ
How do I compute spherical triangle area from excess?
Send positive finite excess E and radius R. With E in radians, area = R²·E by Girard's theorem.
What is spherical excess E?
It is the amount by which a spherical triangle's interior angles exceed π radians: E = A + B + C − π. This tool takes E directly and does not recompute it from angles.
Can I enter the spherical excess in degrees?
Yes. Choose degrees and the excess is converted with π over 180 before it is multiplied by the radius squared. For comparison, an excess of 0.158407 radians is about 9.076053 degrees.
What does the sphere fraction tell me?
It is the excess divided by 4π, the share of the whole sphere surface that the triangle covers. An excess of 0.158407 radians gives a fraction of 0.012606, so you can skip computing the full surface yourself.
Why is an excess of zero refused?
A zero excess describes a flat or collapsed triangle, not a proper spherical one. The excess must be greater than zero and less than 2π, and the radius must be positive.
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/trig/spherical-triangle-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"excess":0.158407,"radius":1}'const res = await fetch("https://api.kit.forhosting.com/trig/spherical-triangle-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"excess": 0.158407,
"radius": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/spherical-triangle-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"excess": 0.158407,
"radius": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/spherical-triangle-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"excess":0.158407,"radius":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"excess":0.158407,"radius":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/spherical-triangle-area", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"excess": 0.158407,
"radius": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.spherical_triangle_area",
"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
There is no language model, no network call inside the solver, and no randomness: the same excess and radius always yield the same JSON.
Girard's theorem: area equals R squared times spherical excess
The response reports the numeric area, echoes the excess in both radians and degrees, and gives sphere_fraction equals E over four pi so you can read coverage as a pure proportion of the full surface without recomputing four pi R squared yourself.
Input contract, units, validation, and response shape
Optional decimals is an integer from zero through twelve that controls half-up rounding of every numeric field and defaults to six. Radius must be strictly positive. After conversion to radians the excess must lie strictly between zero and two pi, the open range for a proper simple spherical triangle; zero excess is planar or degenerate, and two pi or more is outside a single simple triangle. POST a JSON body with two required finite numbers: excess, the spherical excess E of the triangle, and radius, the sphere radius R. Optional excess_unit accepts the strings rad (default) or deg; when deg is set, excess is converted with E times pi over one hundred eighty before Girard's product is formed. Field aliases are accepted in transport: E or spherical_excess for excess, R or sphere_radius for radius, and unit or angle_unit for excess_unit. Missing fields, non-finite values, empty strings, and invalid units fail with invalid_input and are not billed. On success the body includes the echoed excess (in the unit you sent), excess_rad, excess_deg, excess_unit, radius, area, sphere_fraction, formula string area equals R squared times E (Girard), and method girard. Example: excess zero point one five eight four zero seven radians with radius one returns area zero point one five eight four zero seven and sphere_fraction about zero point zero one two six, matching the unit-sphere case where area equals E.
Where an area-from-excess calculator earns its keep
GIS pipelines that already decompose a spherical polygon into triangles may hand each excess to this endpoint and sum the areas without re-deriving angle sums inside every worker.
Integration use cases
Teaching Girard's theorem with stable numbers
Give students a deterministic area-from-excess oracle for unit and non-unit spheres, with clear invalid_input when excess is not positive.
Technical questions
Can I enter excess in degrees?
Yes. Set excess_unit to "deg"; the excess is converted with E·π/180 before area = R²·E is evaluated. The default unit is radians.
Why did I get invalid_input?
Radius must be positive, excess after conversion to radians must lie strictly between 0 and 2π, excess_unit must be rad or deg, and all numbers must be finite.
What does the spherical triangle area calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.