Annulus width from area calculator
A ring with an area of 3π and an inner radius of 1 has an outer radius of 2, so its width is 1. Start from a measured or budgeted area and get the outer edge and the radial thickness in the same unit. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Area
- 50.2654824574
- Inner radius
- 3
- Area
- 50.2654824574
- Outer radius
- 5
- Width
- 2
An annulus is the plane region between two concentric circles—the flat ring you see in washers, pipe walls in cross-section, circular track lanes, and target bullseyes. Its area is the difference of two disk areas: A equals π times (outer radius squared minus inner radius squared). This calculator inverts that identity when you already know the area and the inner radius: it recovers the outer radius as the square root of area over π plus the inner radius squared, then reports the radial width as outer minus inner. Use it for homework checks, manufacturing takeoffs, landscape ring beds, and any workflow that starts from a measured or budgeted area rather than from both radii.
What an annulus is and how width follows from area
In Euclidean plane geometry an annulus is the closed set of points whose distance from a fixed centre lies between an inner radius r and an outer radius R, with R greater than or equal to r. The area of that ring is A equals π(R² minus r²), which is simply the area of the large disk minus the area of the hole.
When a design brief, a material estimate, or a lab measurement hands you A and r but not R, rearrange the identity: R squared equals A over π plus r squared, so R is the non-negative square root of that quantity, and the radial width w is R minus r. That width is the uniform thickness of the ring measured along any radius; it is not the arc length of a sector and not the circumference of either edge.
Zero area is allowed and means a degenerate zero-width ring where outer radius equals inner radius. Because the closed form uses only one division by π, one addition, one square root, and one subtraction, there is no iteration, no mesh, and no dependence on centre coordinates—you do not need the circle equation in general form or a CAD sketch.
Prefer a dedicated sector or arc tool when the region is only a partial angular sweep; prefer volume tools when you need a cylindrical shell or torus rather than a flat ring.
Practical tips, limits, and related searches
Prefer area and radius in consistent units so the returned outer radius and width need no post-conversion; if your source data is diameters only, divide by two before calling. Teaching workflows work well with paired examples: the sixteen-π ring with radii three and five; the three-π ring with radii one and two; zero area to show width zero; and a negative area to demonstrate the hard rejection path that protects automated graders from silent absolute-value bugs.
Prefer this tool when you already know the planar ring area and the hole radius and need the outer radius and radial thickness. Prefer circle area or circumference tools when there is no hole, and cylinder or frustum tools when the solid has height. Round-trip checks are easy: after solving, confirm that π times (outer squared minus inner squared) recovers the input area within floating tolerance, and that width plus inner radius equals outer radius.
What you can do with it
Washer, gasket, and pipe-wall sizing
Turn a budgeted material area and a fixed hole radius into outer radius and ring thickness for manufacturing takeoffs.
Landscape and track ring layouts
Recover the outer edge of a circular planting bed or track lane when total paved area and inner radius are known.
FAQ
What formula does the annulus width calculator use?
From A = π(R² − r²) with known area A and inner radius r, outer radius R = √(A/π + r²) and width w = R − r.
Is zero area allowed?
Yes. Zero area means a degenerate zero-width ring: outer radius equals inner radius and width is zero.
What units should I use?
Any consistent pair: if area is in square metres and inner radius in metres, outer radius and width are in metres.
Is the annulus width from area calculator free?
The browser tool is free and runs locally.
What if I only know the diameters?
Halve the inner diameter before entering it, since the calculator works with radii. The outer radius it returns can be doubled again if you need the outer diameter for a drawing or an order.
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/annulus-width-area \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"area":75.39822368615503,"inner_radius":3}'const res = await fetch("https://api.kit.forhosting.com/geom/annulus-width-area", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"area": 75.39822368615503,
"inner_radius": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/annulus-width-area",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"area": 75.39822368615503,
"inner_radius": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/annulus-width-area", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"area":75.39822368615503,"inner_radius":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"area":75.39822368615503,"inner_radius":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/annulus-width-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": 75.39822368615503,
"inner_radius": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.annulus_width_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
Negative area is rejected with a clear invalid_input error instead of silently taking an absolute value that would hide a data-entry bug.
What an annulus is and how width follows from area
A negative area has no geometric meaning as a measure of planar extent, so this capability rejects it with invalid_input rather than inventing a signed shell. Units are whatever you pass: if area is in square metres and inner radius in metres, outer radius and width come back in metres; the API does not convert square feet to square metres or inches to centimetres for you.
How to call the API and what the response contains
Example: area equal to sixteen π (about 50.265482457437) with inner radius three yields outer radius five and width two, matching the classic concentric pair with R equals five and r equals three. Another check: area equal to three π with inner radius one yields outer radius two and width one. Missing fields likewise fail without billing. POST a JSON body with two required fields: area (aliases A and ring_area) and inner_radius (aliases r, r_inner, and inner_r). Both must be finite numbers; area must be greater than or equal to zero, and inner_radius must be greater than or equal to zero. Numeric strings such as "3" or "12.5" are coerced when they parse as finite numbers; empty strings, non-numeric text, Infinity, and NaN fail with invalid_input naming the field. On success the response echoes area and inner_radius after stable twelve-decimal rounding; returns outer_radius from the square-root formula; returns width as outer_radius minus inner_radius; and includes a formula string plus method set to annulus_width_from_area so clients can log which closed form was used. A negative area—for instance negative ten with any valid inner radius—returns invalid_input with a message that area must not be negative; those validation failures are never charged against prepaid balance. Pricing is a flat $0.002 per successful request with unit item. Use the endpoint from geometry homework pipelines, washer and gasket sizing scripts, irrigation ring-bed planners, packaging and print die cutters, interview-prep sandboxes, or any workflow that needs a reproducible outer radius and ring width without installing a computer-algebra package on every machine.
Practical tips, limits, and related searches
Because the math is elementary, students can recompute R by hand for nice multiples of π and match the API line by line after accounting for floating-point display. This page targets people searching for an annulus width from area calculator, ring width from area, or outer radius given annulus area and inner radius; it is not a full circle-equation solver, not a washer-method solid-of-revolution volume tool, and not a great-circle or geographic ring calculator on the Earth. 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. Pin the rounded JSON fields in tests so Worker and browser goldens stay byte-stable across runtimes.
Integration use cases
Homework checks for annulus area inversion
Give students A and r; compare their hand-computed outer radius and width against a deterministic API or free browser result.
Technical questions
Why are negative areas rejected?
Geometric area is a non-negative measure. Negative input returns invalid_input instead of taking an absolute value that would hide data errors.