Spherical lune area calculator
Find the area of the slice of a sphere's surface between two meridians that meet at the poles: a 90° angle covers exactly one quarter of the sphere and 180° covers half. Angles from 0 to 360 degrees. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Angle
- 90
- Angle radians
- 1.5707963268
- Area
- 78.5398163397
A spherical lune is the region of a sphere’s surface bounded by two great semicircles that meet at a pair of antipodal poles. Those meridians form a dihedral angle, and the lune area is simply that angle’s share of the full sphere surface: one full turn (360 degrees or 2π radians) recovers the classic four pi R squared sphere area, while a right angle recovers one quarter of the sphere. Only two measures are required—a positive sphere radius R and a dihedral angle in degrees between zero and three hundred sixty inclusive. Students meet the identity in spherical-geometry units after sphere surface area; surveyors and cartographers use it when a gores panel or globe strip is described by radius and longitude span rather than a full mesh.
What a spherical lune is and which two measures determine it
The sphere radius R is the distance from center to surface. The dihedral angle α is the angle between the two half-planes that cut those meridians, measured in degrees for this capacity and converted internally to radians as θ equals α times pi over one hundred eighty. Because a full sphere surface is the special case α equals three hundred sixty degrees, the lune area is exactly the fraction α over three hundred sixty of the full sphere area four pi R squared.
Algebraically that is the same as A equals two R squared times θ. When α is ninety degrees the area is one quarter of the sphere; when α is one hundred eighty degrees it is half the sphere; when α is zero the figure degenerates to zero area along a single great semicircle pair. Search intent clusters around phrases such as spherical lune area calculator, area of a spherical lune, dihedral angle sphere surface, spherical digon area, and spherical lune radius and angle.
People want numeric results they can check against an answer key or a globe-panel sketch, not only a geometric argument that the region is determined.
How this calculator validates input and applies the closed-form formulas
Radius must be a strictly positive finite number within a stable numeric range. When both checks pass, the engine converts the angle to radians, multiplies two times R squared times that radian measure, and returns the rounded area together with the recovered inputs and the angle in radians. Edge cases are explicit: missing fields, non-numeric strings, zero or negative radius, and angles outside zero to three hundred sixty all fail before any metric is returned.
Boundary values zero and three hundred sixty are accepted: zero yields area zero, and three hundred sixty recovers the full sphere surface area four pi R squared within floating precision. You may use any consistent unit system for radius; area is expressed in the corresponding square units without conversion inside the engine. Homework keys, contest checkers, and globe-gore worksheets that list only R and dihedral angle therefore share one deterministic contract from classroom to continuous integration.
What you can do with it
Homework and exam spherical-lune drills
Students and graders confirm lune surface area from radius and dihedral angle in degrees, including rejection when the angle lies outside the closed 0–360° interval.
FAQ
Which formula does the calculator use?
Area is 2·R²·θ with θ = angle·π/180 in radians. Equivalently, A = (angle/360)·4·π·R², the angle’s share of the full sphere surface area.
What happens if the angle is outside 0–360 degrees?
A spherical-lune dihedral angle must lie between 0 and 360 degrees inclusive; values outside that range are rejected and those calls are not charged.
Is the spherical lune area calculator free?
The browser tool runs free on your device.
How is a spherical lune different from a circular lune or a spherical wedge?
A spherical lune is sphere surface area between two great semicircles from the same style of radius and dihedral angle.
What is the area of a lune with radius 5 and a 90 degree angle?
It is 78.5398163397 square units, one quarter of the sphere's surface. The angle is first converted to 1.5707963268 radians and then multiplied by two times the radius squared.
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/geom/spherical-lune \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/geom/spherical-lune", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/spherical-lune",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/spherical-lune", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/spherical-lune", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 5,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.spherical_lune",
"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
This capacity applies the closed form A equals two R squared times theta (theta in radians), returns radius, angle, angle in radians, and area as structured JSON, and rejects any angle outside the zero-to-three-hundred-sixty degree interval with an explicit invalid_input error.
What a spherical lune is and which two measures determine it
In Euclidean spherical geometry a spherical lune is the surface region lying between two great semicircles that share a common diameter of the sphere as their endpoints. Delivering radius, angle, angle_radians, area, and a fixed method label as JSON makes the result easy to paste into notes, pin in automated tests, or feed into a larger pipeline without reimplementing pi multiplications and degree-to-radian conversion in every language you teach or deploy. This is distinct from a circular lune on a plane (two disk radii and center distance) and from a spherical wedge (solid volume with the same angle inputs).
How this calculator validates input and applies the closed-form formulas
Send a JSON body with both required fields radius and angle. Angle must be a finite number that lies in the closed interval from zero through three hundred sixty degrees inclusive; if the angle is less than zero or greater than three hundred sixty, the request fails with invalid_input and a message that names the zero-to-three-hundred-sixty bound, because a dihedral angle outside a full turn is not a valid Euclidean spherical lune under this contract. Results use a stable decimal rounding so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate spherical-lune surface area while checking homework, sizing a globe gore opened by a known longitude span, verifying a packaging or map panel from a measured radius and dihedral angle, or confirming a workshop drawing without installing a computer-algebra system. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Do not confuse this tool with a circular lune calculator, which uses two circle radii and center distance on a plane, or with a spherical wedge calculator, which returns solid volume rather than surface area from the same style of radius and dihedral angle inputs. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs require positive radius and an angle between zero and three hundred sixty degrees inclusive; angles outside that range are always rejected with invalid_input, and zero or non-finite radius is likewise refused. Results are fully deterministic: the same radius and angle always yield the same angle_radians and area. Wire the endpoint into spherical-geometry drills, contest answer checkers, globe-panel capacity worksheets that list only R and dihedral angle, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why an angle outside zero to three hundred sixty is rejected.
Integration use cases
Deterministic fixtures for spherical-geometry tests
Pipelines that seed unit tests or interactive sphere demos can call a pure spherical-lune oracle instead of reimplementing 2·R²·θ and degree-to-radian conversion in every language.
Globe gores, map panels, and packaging strips from R and opening angle
Shop or cartography notes that list sphere radius and dihedral (longitude) opening angle recover lune surface area as structured JSON for reports, panel charts, or lightweight CAD helpers.
Technical questions
How do I compute spherical lune area with this API?
Send a JSON body with positive radius and angle (degrees, 0–360) to geom.spherical_lune. The response returns radius, angle, angle_radians, area, and method.