Sector radius from area calculator
Work backwards from a slice's area to its circle: an area of about 19.63 with a 90-degree angle gives a radius of 5, plus an arc of 5π/2 and the full perimeter. Useful for curved floor plans and pie charts. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Area
- 19.6349540849
- Angle
- 90
- Angle unit
- Degrees
- Area
- 19.6349540849
- Angle radians
- 1.5707963268
- Angle degrees
- 90
- Radius
- 5
When you already know the area of a pizza slice, a pie chart wedge, or any circular sector and you know the central angle that cuts it out, the missing geometric unknown is usually the radius of the parent circle. This sector radius from area calculator inverts the classic sector-area formula so you get that radius in one deterministic step. Enter a positive area and a non-zero central angle in degrees or radians; the solver returns the radius, the angle converted both ways, the arc length, and the full perimeter of the sector boundary. Zero angle is rejected because a vanishing wedge cannot enclose a positive area with any finite radius.
Formula: radius from sector area and central angle
A circular sector is the region bounded by two radii and the included arc of a circle. Its area A depends only on the radius r of the circle and the absolute central angle |θ| measured in radians: A equals one half times r squared times |θ|, written A = ½ r² |θ|. Solving for the radius gives the inverse used here: r = sqrt(2A / |θ|).
The absolute value of the angle means a negative central angle (clockwise versus counterclockwise) yields the same geometric radius and the same positive area measures; only a true zero angle is forbidden. Once r is known, arc length follows as r × |θ| and the perimeter of the sector outline (two radii plus the arc) is 2r + arc.
Where finding the radius from the area helps
Architecture and interior design often start from a fixed floor area for a circular bay or a curved seating wedge and a fixed opening angle; recovering the parent radius tells fabricators how large to cut the arc. Data-visualization and reporting tools that rebuild pie or donut charts from published slice areas need the same inverse so labels and radii stay consistent with the numbers already shown to readers.
Exam prep and LMS platforms use a deterministic oracle to mark questions that ask for r given A and θ without trusting student-written square-root code that mishandles degrees versus radians. Failed validation does not consume balance. Keep requests small, pin the rounded radius in tests, and you get reproducible geometry for homework, product pages, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Architecture and curved floor plans
Recover the parent circle radius when a curved bay or seating wedge is specified by area and opening angle for fabrication drawings.
Pie charts and data visualization
Rebuild sector radii from published slice areas and central angles so chart geometry matches the numbers already shown to readers.
Homework and LMS grading
Mark inverse sector problems (find r from A and θ) with a deterministic oracle that rejects zero angle and non-positive area.
FAQ
Does a negative angle change the radius?
No. Geometric measures use the absolute central angle, so clockwise and counterclockwise openings of the same magnitude yield the same radius, arc length, and perimeter.
How is the radius found from the sector area?
Double the area, divide by the angle in radians and take the square root. That reverses the usual rule, where the area is half the radius squared times the angle.
Why is a zero angle not accepted?
A wedge with no opening cannot hold a positive area with any finite radius, so there is no answer. You get a clear message instead of an endless or invented number.
Can I enter the angle in radians?
Yes. Degrees are the default, and you can switch to radians. The result shows the angle both ways, so you do not need a separate converter.
How does this help rebuild a pie chart?
If a report gives each slice's area and angle, the radius tells you how large to draw the chart so its geometry matches the numbers readers already see.
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-radius-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"area":19.634954084936,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/geom/sector-radius-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"area": 19.634954084936,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/sector-radius-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"area": 19.634954084936,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/sector-radius-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"area":19.634954084936,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"area":19.634954084936,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/sector-radius-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
{
"area": 19.634954084936,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.sector_radius_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
Formula: radius from sector area and central angle
When the angle is supplied in degrees, the implementation converts with θ_rad = θ_deg × π / 180 before applying that square root, and it also reports both angle_radians and angle_degrees in the response so you do not need a second converter. This inverse is the natural complement of tools that compute area from radius and angle: here the search intent is radius from area, which is what designers ask when a floor plan or a chart already fixes the wedge area and opening.
Input contract, response fields, and zero-angle errors
If angle is exactly zero the task fails with a clear message that a sector with central angle zero has no finite radius for a positive area—those failures are not billed. Example: area about 19.63 square units and angle 90 degrees yields radius 5, because a quarter-circle of radius 5 has area (π × 25) / 4. POST a JSON object with two required finite numbers: area (positive sector area) and angle (non-zero central angle). Optionally set angle_unit to degrees (the default) or radians. Aliases such as A or sector_area for the area field and central_angle or theta for the angle field are accepted so generators and notebooks can use familiar names. Missing fields, non-finite values, empty strings, and non-numeric types fail with invalid_input naming the problem. If area is zero or negative the request is rejected because a non-positive area does not define a real positive radius under the sector formula. On success the body echoes area, angle, and angle_unit; includes angle_radians and angle_degrees; and returns radius, arc_length, perimeter, and method set to sector_radius_from_area. Pair the numbers with your own drawing code if you need a sketch; this endpoint solves the inverse geometry only and does not emit plots or multi-step prose beyond the structured fields.
Where a sector-radius-from-area API earns its keep
Manufacturing and CNC workflows that nest sector-shaped blanks can call a tiny parse API instead of pulling a full CAD kernel for one algebraic step.
Technical questions
How do I find the radius of a circular sector from its area?
Send a positive area and a non-zero central angle (degrees by default, or radians via angle_unit). The API returns r = sqrt(2A / |θ|) with θ in radians, plus arc length and perimeter.
What happens if the central angle is zero?
The task fails with invalid_input. A sector with zero central angle cannot enclose a positive area with any finite radius, so the request is not billed.
Can I enter the angle in radians?
Yes. Set angle_unit to "radians". The default is degrees. Both angle_radians and angle_degrees appear in every successful response.
What does the sector radius from 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.