Perimeter of a sector
A sector with radius 5 and a 90° angle has a perimeter of 10 + 5π/2: two radii of 5 plus a quarter-circle arc. Enter the radius and central angle to get the arc, the straight sides and the whole boundary. 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
- Two radii
- 10
The perimeter of a sector is the length of the closed pie-slice boundary: one radius out from the center, the circular arc that the central angle subtends, and the other radius back. Given a non-negative radius and a strictly positive central angle in degrees or radians, this tool returns that perimeter together with the two-radii contribution, the arc length alone, both angle representations, and a short formula string so students, fabricators, and automation pipelines share one deterministic answer.
How the perimeter of a sector is defined and computed
A circular sector is the region of a disk bounded by two radii and the shorter or longer arc that those radii cut from the circumference. Its perimeter is not the area of that slice and it is not the arc length by itself. Walk the boundary once: travel from the center along the first radius, follow the circular arc, then return along the second radius.
When the caller supplies the angle in degrees, convert first: θ in radians equals the degree measure times π divided by one hundred eighty. That identity is exact for any finite non-negative radius and any finite positive central angle; you do not need the sector area, the chord, the sagitta, 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 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 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.
Because the path uses only a fixed π conversion, two multiplications, and one addition, it is deterministic, constant time, and free of iterative solvers.
Where a perimeter-of-a-sector 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 they 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-positive 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 user-interface pie charts and level geometry; a deterministic perimeter supports property panels and material estimates without a full mesh library.
Exam-prep generators and learning-management auto-graders need a tiny closed-form checker that rejects a non-positive angle as invalid rather than silently taking an absolute value or treating a zero opening as a line segment.
What you can do with it
Homework and exam checks
Compute the perimeter of a sector as two radii plus arc from radius and central angle, and reject a non-positive 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 zero or negative?
The angle must be a finite positive number; the error message names the field and states that non-positive central angles are rejected.
What does this perimeter-of-a-sector tool cost?
The browser tool is free and runs locally.
Does the perimeter of a sector include both radii?
Yes. The boundary of a pie slice runs out along one radius, around the arc and back along the other radius, so the perimeter is two times the radius plus the arc length. The arc is also shown on its own.
Can I enter the angle in radians?
Yes. Degrees are the default, but you can switch the unit to radians. Either way the result shows the angle in both units, and the arc is the radius times the angle in radians.
What is the perimeter of a half-circle sector?
For a half turn the arc is half the circumference. With a radius of 10 the arc is 10π, so the perimeter is 20 plus 10π: the diameter straight across plus the curved half.
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/math/perimeter-of-a-sector \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/math/perimeter-of-a-sector", {
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/math/perimeter-of-a-sector",
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/math/perimeter-of-a-sector", 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/math/perimeter-of-a-sector", 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": "math.perimeter_of_a_sector",
"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 radius or a non-positive angle fails with invalid_input instead of folding into an absolute value or treating a degenerate slit as a sector. The free browser path and the paid API path run the same pure module, so classroom demos and batch scripts never disagree on two radii plus arc.
How the perimeter of a sector is defined and computed
If r is the radius and θ is the central angle expressed in radians, the curved piece has length r times θ and the two straight pieces add 2r, so the perimeter of a sector is P = 2r + rθ, which factors as r times the quantity two plus θ. Huge magnitudes can still overflow double range; when intermediates are not 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
Length units 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. Radius must be a finite number that is not negative; a strictly negative radius is rejected with invalid_input and a message that names the field. Angle must be a finite positive number; zero and negative openings are non-positive and fail with invalid_input so a degenerate slit is never silently treated as a sector. 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, two_radii equal to twice the radius, arc_length equal to r times θ in radians, perimeter equal to two radii plus the arc, and formula set to P = 2r + r·θ (θ in radians). Example: radius five and angle ninety degrees yield two_radii ten, arc length about seven point eight five four, and perimeter about seventeen point eight five four.
Where a perimeter-of-a-sector 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 phrase that matches how people ask for the perimeter of a sector from radius and angle at $0.002 per successful API call, distinct from area-first landings that answer a different primary keyword.
Integration use cases
Sector metrics in CAD and games
Turn radius and angle into perimeter, two radii, and arc length for property panels, material estimates, and deterministic fixtures.
Technical questions
What formula gives the perimeter of a sector?
P = 2r + r·θ with θ in radians (degrees are converted first). The response also returns two_radii = 2r, arc_length = r·θ, both angle units, and the formula string.
Does the perimeter include the two radii?
Yes. two_radii is the pair of straight sides, arc_length is the curved part alone, and perimeter is their sum. 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.