Lune of Hippocrates calculator
With a leg of 6, the lune of Hippocrates has an area of 18, exactly the same as the isosceles right triangle it is built on. Enter one leg to get the lune, the triangle, the hypotenuse and the semicircle on the leg. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Leg
- 6
- Hypotenuse
- 8.4852813742
- Triangle area
- 18
- Lune area
- 18
- Semicircle on leg area
- 14.1371669412
The lune of Hippocrates is one of the earliest exact area results in classical Greek geometry: a crescent formed by two circular arcs has area equal to a right triangle built on the same figure. For the isosceles right triangle with leg length a, that equality is simply L equals a squared over two—the same formula as the triangle area itself. Students, teachers, and history-of-math writers need a reliable number without redrawing semicircles by hand or fighting floating-point noise in a notebook. This lune of Hippocrates calculator accepts a single positive leg, returns the hypotenuse a√2, the triangle area, the semicircle area on that leg, and the lune area that matches the triangle, with a short formula string for display.
What the lune of Hippocrates is and why its area equals the triangle
Hippocrates of Chios (fifth century BCE) studied regions bounded by circular arcs long before a general theory of circle quadrature existed. In the classical construction you start with an isosceles right triangle whose equal legs have length a and whose hypotenuse is a times the square root of two. Draw the semicircle that has one leg as diameter; its area is one half pi times (a over two) squared, which simplifies to pi a squared over eight.
The same triangle is inscribed in a larger circle whose diameter is the hypotenuse. Hippocrates proved that this crescent has exactly the same area as the right triangle: both equal a squared over two. The proof is elementary once you compare segments and right-triangle areas: the curved excesses cancel so that the remaining lune matches the polygonal triangle.
That equality is the educational heart of this capability. You do not need a numerical integration of two arcs; you need a positive leg and the closed form L equals T equals leg squared over two. The hypotenuse and the semicircle-on-leg area are included so lesson plans and worksheets can show intermediate construction quantities without recomputing them in a second tool.
Who uses the lune of Hippocrates calculator
Classrooms covering the history of Greek geometry use a lune of Hippocrates calculator so learners can plug in a leg from a textbook figure and verify that the crescent area really equals the triangle without messy compass constructions on paper. Exam-prep and LMS plugins need a deterministic oracle that rejects non-positive legs instead of returning a silent zero that a buggy grader might treat as success. Museum education teams and MOOC authors often want intermediate quantities—hypotenuse, semicircle area on the leg—alongside the famous equality for captions and progressive disclosure.
Keep requests small, pin the rounded fields in tests, and you get reproducible Hippocrates areas for homework, CMS embeds, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom history-of-geometry checks
Let students enter the leg from a textbook lune figure and confirm that lune area equals triangle area leg²/2 with matching intermediate construction values.
LMS and exam auto-grading
Mark lune-of-Hippocrates items with a deterministic oracle that rejects non-positive legs and never invents a crescent area on invalid input.
FAQ
How do I compute the lune of Hippocrates area?
Send a finite positive leg of the isosceles right triangle in the classical construction.
Why does the lune area equal the triangle area?
That is Hippocrates' theorem for this construction: after comparing circular segments and the isosceles right triangle, the crescent simplifies to leg²/2, the same as the triangle.
What happens if the leg is zero or negative?
A leg must be a positive length; zero and negative values are rejected and are not billed.
Is the triangle always isosceles right?
Yes for this capability. The classical single-leg lune of Hippocrates uses an isosceles right triangle (legs equal, hypotenuse leg·√2). Pass one leg only.
Which crescent does the lune of Hippocrates measure?
It is the region inside the small semicircle drawn on one leg but outside the arc of the larger circle whose diameter is the hypotenuse. Both arcs end at the two ends of that leg, which gives the crescent its shape.
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/lune-hippocrates \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"leg":6}'const res = await fetch("https://api.kit.forhosting.com/geom/lune-hippocrates", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"leg": 6
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/lune-hippocrates",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"leg": 6
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/lune-hippocrates", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"leg":6}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"leg":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/lune-hippocrates", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"leg": 6
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.lune_hippocrates",
"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
Non-positive legs fail with invalid_input so homework tools and APIs never invent a crescent on a zero or negative side.
What the lune of Hippocrates is and why its area equals the triangle
The lune—or crescent—is the region that lies inside the small semicircle but outside the arc of the larger circle between the endpoints of that leg. Reporting both the triangle area and the lune area side by side makes the theorem visible in the API response, and the boolean equal flag confirms they match after stable twelve-decimal rounding.
Input contract, response fields, and validation rules
The leg must be strictly positive: zero and negative lengths are rejected because a right-triangle leg and a semicircle diameter are geometric lengths, not signed displacements. Example: leg equals six yields triangle and lune areas of eighteen, hypotenuse about eight point four eight five, and a semicircle-on-leg area of about fourteen point one three seven. Failed validation is not billed. POST a JSON body with one required finite number: leg (aliases a, side, and leg_length are accepted). The value may be a JSON number or a numeric string; missing fields, empty strings, non-numeric text, and non-finite values fail with invalid_input naming the field. On success the response echoes the leg, reports hypotenuse as leg times the square root of two, triangle_area as leg squared over two, lune_area equal to that same quantity, semicircle_on_leg_area as pi times leg squared over eight, equal set to true, plus formula and method strings for UI labels. Pair the numbers with your own diagram if you need a sketch; this endpoint is pure analytic geometry and does not emit SVG or plot data. Successful calls use the published $0.002 base price on the API path; the browser tool remains free for interactive checks.
Where a lune of Hippocrates API earns its keep
Engineering notebooks occasionally cite the lune as a sanity check when teaching circular segments; calling a tiny parse API avoids pulling a computer-algebra system for one closed form.
Integration use cases
Educational content and museum demos
Power live widgets on articles about Greek quadrature with the same pure result the API returns for automated captions and worksheets.
Technical questions
What does the lune of Hippocrates calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.