ForHosting KIT · Math

Arc length calculator

With a radius of 10 and a 90° central angle, the arc measures 5π and the sector covers 25π square units, exactly a quarter of the circle. Enter the angle in degrees or radians; the full circumference is shown too. Free in your browser.

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Worked example

If you enter

Radius: 10
Angle: 90

You get

Angle unit: Degrees
Angle radians: 1.5707963268
Angle degrees: 90
Arc length: 15.7079632679

Finding the length of a circular arc and the area of the pie-slice sector it bounds is a staple of geometry homework, mechanical design, surveying, and game physics. Given a positive radius and a central angle—the angle at the circle's centre subtended by the arc—you convert the angle to radians if needed, multiply radius by that angle for arc length, and take half the product of radius squared and the same angle for sector area. Results include both arc length and sector area, plus supporting fields such as angle in both units, full circumference, full-disk area, and the fraction of the circle swept.

What arc length and sector area mean

On a circle of radius r, a central angle θ measured in radians sweeps an arc whose length is simply r times θ. That identity is the definition of radian measure: an angle of one radian subtends an arc equal in length to the radius. When the angle is given in degrees, first convert with θ_rad equals θ_deg times π divided by one hundred eighty, then apply the same product.

The circular sector is the region bounded by the two radii and the arc—the classic pie slice. Its area is one half r squared times θ in radians, which is also the fraction θ over two π of the full disk area π r squared. A ninety-degree central angle is one quarter of the circle: for r equals ten, arc length is five π and sector area is twenty-five π.

A full turn of three hundred sixty degrees or two π radians recovers the circumference two π r and the full area π r squared. Zero angle yields zero arc length and zero sector area, which is allowed; a negative angle does not correspond to a geometric central measure in this contract and is rejected. The formulas are elementary multiplications after at most one conversion, so there is no iteration, no sampling, no dependence on wall-clock time, and no need for chord length, sagitta, or coordinates in the input.

What you can do with it

Mechanical and sheet-metal arc takeoff

Turn a bend radius and swept angle into arc length for material estimation, or sector area for plate cutouts, without a drawing office calculator.

Game, UI, and chart pie-slice geometry

Compute arc lengths and sector areas for progress rings, pie charts, and circular progress indicators from radius and central angle in degrees or radians.

FAQ

What is the formula for arc length?

With radius r and central angle θ in radians, arc length is r·θ. If θ is in degrees, convert first: θ_rad = θ_deg·π/180, then multiply by r.

What is the formula for sector area?

Sector area is ½·r²·θ with θ in radians, which equals the fraction θ/(2π) of the full circle area πr².

Why are negative central angles rejected?

A central angle is a non-negative measure of how much of the circle is swept.

Can I enter the angle in radians?

The default is degrees for classroom and construction workflows.

Is the arc length calculator free?

The browser tool is free and runs locally.

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

POSThttps://api.kit.forhosting.com/geom/arc-length

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/arc-length \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"radius":10,"angle":90}'

const res = await fetch("https://api.kit.forhosting.com/geom/arc-length", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "radius": 10,
    "angle": 90
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/geom/arc-length",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "radius": 10,
    "angle": 90
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/arc-length", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"radius":10,"angle":90}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"radius":10,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/arc-length", 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": 10,
  "angle": 90
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "geom.arc_length",
  "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

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Limits

`max_radius`	1000000000000
`min_radius`	1e-15
`max_angle`	1000000000000

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

This capability packages that closed form as deterministic JSON: no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path. A negative central angle is rejected with invalid_input rather than silently taking an absolute value, so scripts and grading pipelines see an honest error when the measure is invalid.

How to call the API and what comes back

Use it from geometry homework pipelines, CNC and sheet-metal takeoff scripts, game-level tooling, interview-prep sandboxes, or any workflow that needs reproducible arc and sector numbers without installing a computer-algebra package on every machine. Send a JSON body with radius and angle. Field radius is the positive circle radius; aliases r and R are accepted. Field angle is the non-negative central angle; aliases central_angle, theta, and the θ character map to the same meaning. Optional angle_unit is degrees by default or radians when you pass radians (aliases unit and angle_mode). On success the response echoes radius, angle, and angle_unit; adds angle_radians and angle_degrees for convenience; returns arc_length and sector_area as the primary results; and includes circumference, full_circle_area, fraction_of_circle, method set to arc_sector, and a short formula block documenting the closed forms. Failed validation—missing a required field, a non-finite number, a non-positive radius, a negative central angle, an unknown angle unit, or a value outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, limits, and related searches

Angle may be zero or larger than a full turn when you intentionally model multi-revolution sweeps; only negative values are rejected. When teaching, prefer paired examples: radius ten and ninety degrees for quarter-circle checks against five π and twenty-five π; radius one and π radians for a semicircle arc of length π and sector area π over two; and a negative angle to demonstrate the hard rejection path. Prefer this tool when you already know the central angle and radius and need the arc and the pie-slice area. Round-trip checks are easy: after solving, confirm that arc length over circumference equals sector area over full disk area within floating tolerance, and that both equal θ over two π when θ is in radians. Keep radius inside the published positive range up to one trillion so intermediate products stay finite in IEEE double precision on the Worker and in the browser. Units of length are whatever you pass—metres, inches, or game units—as long as you interpret arc length in that same unit; the API does not convert linear units for you. Because the formulas only need r and θ, students can recompute by hand for nice fractions of a circle and match the API line by line. This landing targets people searching for an arc length calculator or a sector area from radius and central angle tool; it is not a full circle-equation solver from general form coefficients—use the dedicated circle equation capability when you need standard form, general form, and area from centre and radius without an arc angle. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract.

Integration use cases

Homework checks for arc length and sector area

Give students a radius and central angle; compare their hand-computed arc length and sector area against a deterministic API or free browser result.

Read the full KIT documentation →

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