Circular sector perimeter calculator
Measure the full outline of a pie slice: a radius of 5 at 90 degrees gives an arc of 5π/2 and a perimeter of 10 plus that arc, since both straight sides count. Handy for trim, edging and cut paths. 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
- Arc length
- 7.853981634
A circular sector is the pie-slice region bounded by two radii and the arc between them. Its perimeter is the length of that closed boundary: walk out along one radius, around the arc, and back along the other radius. Given a positive radius and a non-negative central angle in degrees or radians, this circular sector perimeter calculator returns the perimeter together with the arc length, both angle representations, and a short formula string so students, fabricators, and automation pipelines share one deterministic answer.
How circular sector perimeter is defined and computed
Label the distance from the circle center to the circumference as the radius r, and the central angle between the two bounding radii as θ. Convert θ to radians when the caller supplies degrees: θ_rad equals θ_deg times π over one hundred eighty. The arc length alone is r times θ_rad.
The two straight sides contribute two times r. The full perimeter of the sector region is therefore P equals two r plus r times θ_rad, which factors as r times the quantity two plus θ_rad. That identity is exact for any positive finite radius and any non-negative finite central angle; you do not need the sector area, the chord length, or the circular segment area unless you are answering a different question.
Classic checks include radius five and a right angle of ninety degrees, which yields arc length five times π over two and perimeter ten plus that arc; radius ten and a half-turn of one hundred eighty degrees, which yields arc length ten times π and perimeter twenty plus that arc; and a full turn of three hundred sixty degrees, where the perimeter becomes two r plus the full circumference two π r.
Where a circular sector perimeter calculator earns its keep
Secondary-school and college plane-geometry courses constantly ask for the perimeter of a sector given radius and central angle, and often distinguish the arc length alone from the full perimeter that includes both radii on the same problem set; a trusted calculator with a strict non-negative angle gate speeds homework grading and removes spreadsheet mistakes where someone uses only the arc or forgets one of the two radii.
Architecture, metalwork, and sign fabrication size pie-slice panels, arched openings, and decorative sectors from outer radius and opening angle; perimeter informs trim, gasket, edging, and laser cut path length while area tools answer fill separately. Landscape and civil plans place sector-shaped beds, plazas, and paved wedges; perimeter drives curb, railing, and border estimates once the design freezes radius and sweep. Game, graphics, and CAD libraries store sector primitives for UI pie charts and level geometry; deterministic perimeter supports property panels and material estimates without a full mesh library.
Exam-prep generators and LMS auto-graders need a tiny closed-form checker that rejects negative angles as invalid rather than silently taking an absolute value.
What you can do with it
Homework and exam checks
Compute sector perimeter as two radii plus arc from radius and central angle, and reject a negative angle with a clear error.
Trim, edging, and cut-path length
From design radius and sweep angle, recover total boundary length for gaskets, rails, and laser paths around a pie-slice panel.
FAQ
What happens if the central angle is negative?
The angle must be a finite non-negative number; the error message names the field and states that the angle must not be negative.
What does the circular sector perimeter calculator cost?
The browser tool is free and runs locally.
Is the sector perimeter the same as the arc length?
No. The arc is only the curved part; the perimeter adds the two straight radii. Forgetting one of them, or using the arc alone, is the most common slip with sectors.
What is the perimeter of a half-turn sector with radius 10?
The arc is 10π and the two radii add 20, so the perimeter is 20 plus 10π. The same rule works for any opening angle you enter.
Does the calculator convert length units?
No. The perimeter comes out in the unit you used for the radius, so centimetres give centimetres. It also does not draw the sector or give its area.
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-perimeter \
-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-perimeter", {
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-perimeter",
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-perimeter", 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-perimeter", 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_perimeter",
"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
A negative central angle fails with invalid_input instead of silently folding into an absolute value, matching the product contract for this perimeter-focused landing. The free browser tool and the paid API path run the same pure module, so classroom demos and batch scripts never disagree on two radii plus arc.
How circular sector perimeter is defined and computed
Because the path uses only multiplication, addition, a fixed π conversion, and ordinary IEEE-754 arithmetic, it is deterministic, O(1), and free of iterative solvers. Huge magnitudes can still overflow 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 elementary geometry textbooks and keeps automated graders aligned with hand computation on the same radius and angle.
Input contract, response shape, and error behaviour
Radius must be a finite number strictly greater than zero. Example: radius five and angle ninety degrees yield arc length about seven point eight five four and perimeter about seventeen point eight five four. Units of length are caller-defined: if you send centimetres you get centimetres of perimeter; the calculator does not convert length units, draw the sector, or return area. Pair the numeric result with your own unit labels and drawings. Keep requests small, pin the rounded perimeter in tests, and you get reproducible two-radii-plus-arc sums for homework, LMS plugins, and edge workers without standing up a computer-algebra system. POST a JSON body with two required fields: radius for the circle radius and angle for the central angle of the sector. Angle must be a finite non-negative number; a strictly negative angle is rejected with invalid_input and a message that names the field and the non-negative rule. Optional angle_unit defaults to degrees and may be set to radians; aliases unit and angle_mode are accepted, as are short aliases r and radius_length for radius and central_angle or theta for angle. Fields may arrive as JSON numbers or as numeric strings; empty strings, missing fields, Infinity, and NaN fail naming the problem. On success the body includes radius and angle echoed with twelve-decimal rounding, angle_unit, angle_radians and angle_degrees for both unit views, arc_length, perimeter equal to two radii plus the arc, formula set to P = 2r + r·θ (θ in radians), and method set to sector_perimeter_two_radii_plus_arc for logging and multi-solver UIs.
Where a circular sector perimeter calculator earns its keep
In all of those settings the value is not exotic mathematics: it is a stable contract, clear errors, and a search intent that matches how people query for sector perimeter from radius and angle at $0.002 per successful API call, distinct from area-only landings that answer a different primary keyword.
Integration use cases
Sector metrics in CAD and games
Turn radius and angle into perimeter and arc length for property panels, material estimates, and deterministic fixtures.
Technical questions
What formula does this circular sector perimeter calculator use?
P = 2r + r·θ with θ in radians (degrees are converted first). The response also returns arc_length = r·θ, both angle units, formula, and method: "sector_perimeter_two_radii_plus_arc".
Does the perimeter include the two radii?
Yes. arc_length is the curved part alone; perimeter adds both straight radii for 2r + arc overall. That is the closed boundary of the sector region.
Can I send the angle in radians instead of degrees?
Yes. Set angle_unit to "radians" (default is "degrees"). Aliases unit and angle_mode are accepted.