Annulus sector area calculator
A 90° slice of a ring with outer radius 10 and inner radius 4 covers 65.973445725386 square units, a quarter of the full ring. Enter the angle in degrees or radians; an inner radius larger than the outer one is refused. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Outer radius
- 10
- Inner radius
- 4
- Angle
- 90
- Angle unit
- Degrees
- Angle radians
- 1.5707963268
- Angle degrees
- 90
- Area
- 65.9734457254
An annulus sector is the curved ring-slice bounded by two concentric circular arcs and two radial edges. Given a positive outer radius, a non-negative inner radius that does not exceed the outer, and a central angle in degrees or radians, this calculator returns the sector area of the annular region.
How annulus sector area is computed
Convert the central angle to radians when it arrives in degrees: θ_rad equals angle times π over one hundred eighty. The geometric area of the annulus sector is one half times the absolute value of that angle in radians times the difference of the squared radii, outer radius squared minus inner radius squared.
That closed form is identical to the familiar classroom expression (θ° divided by three hundred sixty) times π times (R squared minus r squared), and it equals the difference between two circular sector areas that share the same central angle. Using the absolute value keeps the reported area non-negative if a signed angle is flipped for orientation, while the response still echoes the original signed angle in both degrees and radians for audit trails.
Classic checks include outer radius ten, inner radius four, and a ninety-degree central angle, which yields area one half times π over two times (one hundred minus sixteen), or (eighty-four π) over four. Because the path uses only multiplication, subtraction of squares, division by constants, and absolute value after the π conversion, it is deterministic, O(1), and free of iterative solvers.
Where annulus sector calculations show up in real work
Secondary-school and college geometry courses constantly ask students for the area of a ring slice or washer sector given two radii and a central angle; a trusted calculator with a strict r-not-exceeding-R gate speeds homework grading and removes spreadsheet mistakes where someone multiplies by π without converting degrees or forgets to subtract the inner disk sector.
Manufacturing and drafting workflows use annulus sectors for gasket arcs, flange cutouts, and cam ring lobes; having a shared numeric oracle keeps shop notes and office scripts aligned on the same half-theta times R-squared-minus-r-squared path. Game, graphics, and UI libraries often store progress rings, donut-chart slices, and annular cooldown wedges as polar ring sectors; area feeds layout math and collision proxies without a full computational-geometry stack.
If you later need only a full circular sector from a single radius, a complete annulus area without an angle, or spherical-sector volume, treat those as separate search intents rather than overloading this annulus-sector landing.
What you can do with it
Homework and exam checks
Compute annulus sector area from two radii and degrees or radians, and reject inner greater than outer with a clear error.
Gaskets and flange cutouts
From positive outer and non-negative inner radii plus a central angle, recover the ring-slice area for drafting and shop notes.
FAQ
What formula does this calculator use?
With θ in radians: area = ½ |θ| (R² − r²). Degrees are converted with θ_rad = angle × π / 180. The response includes method: "annulus_sector_half_theta_R2_minus_r2".
What happens if the inner radius exceeds the outer radius?
An annulus sector requires r ≤ R; the error message names the ordering constraint.
Does a full-ring sector (360 degrees) work?
Yes. An angle of 360 degrees (or 2π radians) yields the full annulus area π (R² − r²).
What does it cost?
The browser tool is free and runs locally.
Can I enter the angle in radians?
Yes. Degrees are the default, but you can switch the unit to radians and type the same angle that way. The result shows the angle in both units, so you can check the conversion.
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/annulus-sector-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"outer_radius":10,"inner_radius":4,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/geom/annulus-sector-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"outer_radius": 10,
"inner_radius": 4,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/annulus-sector-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"outer_radius": 10,
"inner_radius": 4,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/annulus-sector-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"outer_radius":10,"inner_radius":4,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"outer_radius":10,"inner_radius":4,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/annulus-sector-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
{
"outer_radius": 10,
"inner_radius": 4,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.annulus_sector_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
When the inner radius exceeds the outer radius the request fails with a clear invalid_input error instead of a silent negative area that looks like a successful solve. Angles may be given in degrees by default or in radians when angle_unit is set. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the numbers you provide.
How annulus sector area is computed
Huge radius or angle magnitudes can still overflow IEEE-754 double range; when the intermediate area is non-finite the request fails with a clear overflow message instead of returning Infinity in JSON. That closed form matches analytic geometry textbooks and keeps automated graders aligned with hand computation on washer and ring-slice problems.
What you send and what you get back
Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing π conversion in every language. Provide three required finite numbers: outer_radius must be strictly positive, inner_radius must be non-negative and must not exceed the outer radius, and angle is the central angle of the sector. Optionally set angle_unit to degrees (the default) or radians so the same numeric angle can be interpreted either way. Missing fields, empty strings, non-numeric text, NaN, Infinity, non-positive outer radius, negative inner radius, and an inner radius that exceeds the outer are rejected with invalid_input and are not charged. On success the JSON response returns outer_radius and inner_radius, the original angle, the resolved angle_unit, angle_radians and angle_degrees for dual display, area, and method set to annulus_sector_half_theta_R2_minus_r2 so logs and teaching UIs can document the algorithm. Example checks: outer ten, inner four, and angle ninety degrees yield a quarter-annulus area; outer five, inner zero, and angle π radians with angle_unit radians recover a half-disk of radius five; inner greater than outer fails naming the ordering constraint. API pricing is the published base of $0.002 per successful item.
Where annulus sector calculations show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Integration use cases
Donut charts and progress rings
Turn outer radius, inner radius, and central angle into sector area for UI rings, cooldown wedges, and annular pie slices in browser or API.
Technical questions
Can I enter the angle in radians?
Yes. Set angle_unit to "radians" (default is "degrees"). The response always returns both angle_radians and angle_degrees.