Ellipse perimeter calculator
Get the distance around an ellipse from its two semi-axes with Ramanujan's first approximation: semi-axes of 5 and 3 give about 25.526986393759, using one square root and no integrals. A circle comes out exactly. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 5
- B
- 3
- Perimeter
- 25.5269863938
The exact perimeter of an ellipse is an elliptic integral with no elementary closed form, so engineers, teachers, and spreadsheet authors rely on accurate algebraic approximations instead of numerical quadrature. Ramanujan's first formula uses only addition, multiplication, and one square root on the two semi-axes and is already excellent for modest eccentricity while remaining exact for the circle.
What Ramanujan's first ellipse perimeter means and when to use it
An ellipse with semi-axes a and b is the set of points whose distances to two foci sum to a constant. Its area is elementary: π times a times b. Its perimeter is not elementary: the complete elliptic integral of the second kind has no closed form in radicals and elementary functions.
For hand calculation, firmware, and low-latency APIs you therefore want a short algebraic stand-in. Srinivasa Ramanujan published several such stand-ins; the first and simplest is P approximately equal to π times the quantity three times (a plus b) minus the square root of (three a plus b) times (a plus three b). When a equals b the expression collapses exactly to two π a, the circumference of a circle, so the approximation introduces no circle error.
When the axes differ moderately the relative error stays small compared with typical drafting and material tolerances; for extremely flat ellipses the second Ramanujan formula or series methods improve accuracy, but those are a different search intent and a different landing. Keeping the contract to two inputs and one primary output makes the calculator ideal as a fast check beside a diagram, a two-field form, or a one-line step in a larger geometry pipeline that already chose Ramanujan I for its simplicity and exact circle limit.
What you enter and what comes back
Each must be a finite non-negative number. Example: semi-axes five and three yield a perimeter near twenty-five point five three, matching Ramanujan I rather than a crude average-radius estimate. Units are caller-defined: if you send centimetres you get centimetres of perimeter; if you send metres you get metres.
The calculator does not convert units, prove that a and b came from a measured figure, draw the ellipse, or switch automatically to Ramanujan II. 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-axis Ramanujan-I sums for homework, LMS plugins, and edge workers without standing up a computer-algebra system.
Where an ellipse perimeter Ramanujan calculator earns its keep
Geometry homework and entrance exams often ask for the approximate circumference of an ellipse given only the two semi-axes, sometimes after students have already computed a and b from a major-axis length and a minor-axis length. Learners sometimes average a and b and multiply by two π, which is a coarse and biased estimate; a deterministic oracle that always applies Ramanujan I catches those shortcuts before a tutor spends a session on elliptic integrals.
Craft, roofing, CNC, and track-layout workflows that need the total edge length of an elliptical path for material estimates freeze a and b from design drawings and want a closed-form number without importing numerical libraries. Landscape and event planners who lay elliptical stages, garden beds, or running tracks convert measured semi-axes into total border or curb length without opening a full CAD suite.
Exam-prep generators and LMS auto-graders need a tiny closed-form checker that rejects negative semi-axes as invalid rather than silently returning a negative perimeter that a buggy grader might accept.
What you can do with it
Classroom and homework checks
Students enter semi-axes a and b and verify the Ramanujan I perimeter, catching crude 2π((a+b)/2) shortcuts and sign mistakes on the nested radicals.
Craft, track, and material estimates
Convert frozen design semi-axes into total elliptical border length for frames, curb runs, laser paths, or stock cut lists without elliptic-integral libraries.
LMS and exam auto-grading
Mark ellipse-perimeter items with a deterministic Ramanujan-I oracle that rejects any negative semi-axis and never invents area or eccentricity when only the perimeter is required.
FAQ
How do I calculate ellipse perimeter with Ramanujan's first approximation?
Send two finite non-negative semi-axes a and b.
What happens if any semi-axis is negative?
An ellipse cannot have a negative semi-axis, and invalid input is not billed.
Is this the same as Ramanujan's second approximation?
Ramanujan II uses h=((a−b)/(a+b))² and a different radical form; use a dedicated second-approximation or multi-metric ellipse tool if you need that formula.
Is Ramanujan's first formula exact for a circle?
Yes. When both semi-axes are equal, the expression collapses to exactly two pi times the radius, the circumference of a circle, so in that case it introduces no error at all.
When is a more accurate method worth it?
For very flat ellipses. With moderate stretching the first formula stays well within normal drafting tolerances; for extreme shapes, Ramanujan's second formula or a series method gets closer.
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/ellipse-perimeter-ramanujan \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":5,"b":3}'const res = await fetch("https://api.kit.forhosting.com/geom/ellipse-perimeter-ramanujan", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 5,
"b": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/ellipse-perimeter-ramanujan",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 5,
"b": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/ellipse-perimeter-ramanujan", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":5,"b":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":5,"b":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/ellipse-perimeter-ramanujan", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 5,
"b": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.ellipse_perimeter_ramanujan",
"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
This ellipse perimeter Ramanujan calculator accepts finite non-negative semi-axes a and b, returns the approximate circumference together with the echoed inputs and a short formula string, and fails with invalid_input when any semi-axis is negative so callers never bill a geometrically impossible figure.
What Ramanujan's first ellipse perimeter means and when to use it
This endpoint intentionally answers only the Ramanujan-I boundary question from two semi-axes. It does not return area, eccentricity, foci, or a numerical elliptic-integral residual.
Input contract, response shape, and error behaviour
POST a JSON body with two required fields: a and b for the semi-axis lengths of the ellipse. Fields may arrive as JSON numbers or as numeric strings; empty strings, missing fields, Infinity, and NaN fail with invalid_input naming the problem. Aliases semi_axis_a, semiAxisA, and A map to a; semi_axis_b, semiAxisB, and B map to b so clients that already use longer geometry names do not need a rename layer. On success the body includes the echoed a and b (each rounded to twelve decimal places for stable goldens), perimeter equal to π times three times (a plus b) minus the square root of (three a plus b) times (a plus three b) with the same rounding, formula equal to a short Ramanujan-I string, and method equal to ramanujan_i for logging and multi-solver UIs.
Where an ellipse perimeter Ramanujan calculator earns its keep
Engineering notebooks and spreadsheet replacements can call this endpoint as a two-field step when a larger script already holds semi-axes for several figures and only the Ramanujan-I perimeter case needs a dedicated, SEO-clear name distinct from full ellipse metric suites or second-approximation tools. 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 actually query for ellipse perimeter Ramanujan approximation from two semi-axes at $0.002 per successful API call.
Technical questions
Which units does the ellipse perimeter Ramanujan calculator use?
Any consistent length unit. If a and b are in metres, perimeter is in metres; the API does not convert units or attach unit labels.
What does the ellipse perimeter Ramanujan calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.