Circular sector area and perimeter
Give the radius and central angle of a pie-slice sector and get its area: a quarter circle of radius 5 at 90 degrees covers 25π/4, with an arc of 5π/2 and a perimeter of 10 plus that arc. Degrees or radians both work. 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 unit
- Degrees
- Angle radians
- 1.5707963268
- Angle degrees
- 90
- Area
- 19.6349540849
- Arc length
- 7.853981634
A circular sector is the pie-slice region bounded by two radii and the included arc. Given a positive radius and a central angle in degrees or radians, this calculator returns the sector area, the arc length, and the full perimeter of the sector boundary (two radii plus the arc).
How sector area and perimeter are 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 sector is one half times radius squared times the absolute value of that angle in radians, which is the same as the familiar classroom form (θ° divided by three hundred sixty) times π r squared. Arc length is radius times absolute θ_rad, and the perimeter of the sector as a closed path is two radii plus that arc.
Using the absolute value keeps area, arc length, and perimeter 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 a quarter-circle of radius five with a ninety-degree central angle, which yields area (25π)/4, arc length (5π)/2, and perimeter ten plus that arc. Because the path uses only multiplication, division by constants, and absolute value after the π conversion, it is deterministic, O(1), and free of iterative solvers.
Where circular sector calculations show up in real work
Secondary-school and college geometry courses constantly ask students for the area of a sector given radius and central angle; a trusted calculator with a strict positive-radius gate speeds homework grading and removes spreadsheet mistakes where someone multiplies by π without converting degrees. Manufacturing and drafting workflows use sectors for cam lobes, pie-slice cutouts, and gasket arcs; having a shared numeric oracle keeps shop notes and office scripts aligned on the same half-r-squared-theta path.
Game, graphics, and UI libraries often store pie charts, skill cooldowns, and cone attacks as polar wedges; area and perimeter feed collision proxies, progress rings, and layout math without a full computational-geometry stack. If you later need full disk area only, segment area from chord height, or annular sector (ring slice) formulas, treat those as separate search intents rather than overloading this circular-sector landing.
What you can do with it
Homework and exam checks
Compute sector area and perimeter from radius and degrees or radians, and reject non-positive radius with a clear error.
Drafting and cam profiles
From a positive radius and central angle, recover area and boundary length for cutouts, pie charts, and gasket arcs.
FAQ
What formulas does this calculator use?
Degrees are converted with θ_rad = angle × π / 180.
What happens if the radius is zero or negative?
A sector needs a positive radius; the error message names the field.
Does a full circle work?
Yes. An angle of 360 degrees (or 2π radians) yields the full disk area π r², arc length 2π r, and perimeter 2r + 2π r.
What does it cost?
The browser tool is free and runs locally.
Do I have to convert degrees to radians myself?
No. Enter the angle in degrees, the default, or switch to radians, and the conversion is done for you. Forgetting that step is a classic cause of wrong sector areas.
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/sector-area \
-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/sector-area", {
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/sector-area",
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/sector-area", 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/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
{
"radius": 5,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.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
Non-positive radius values are rejected with a clear invalid_input error instead of a silent zero 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 sector area and perimeter are computed
Huge radius or angle magnitudes can still overflow IEEE-754 double range; when intermediates are 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.
What you send and what you get back
Provide two required finite numbers: radius must be strictly positive, and angle is the central angle of the sector. 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. 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, zero radius, and negative radius are rejected with invalid_input and are not charged. On success the JSON response returns radius and the original angle, the resolved angle_unit, angle_radians and angle_degrees for dual display, area, arc_length, perimeter, and method set to sector_half_r2_theta so logs and teaching UIs can document the algorithm. Example checks: radius five and angle ninety degrees yield a quarter-disk area; radius one and angle π radians with angle_unit radians yield a semicircle; radius zero or a missing angle fails naming the problem. API pricing is the published base of $0.002 per successful item.
Where circular 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
Game and UI polar wedges
Turn a radius and central angle into area and arc length for pie charts, cooldown rings, and cone proxies 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.