Intersecting chords calculator
With AP of 3, PB of 4 and CP of 2, the missing PD is 6, because both chords share the product 12. Enter any three of the four segments of two crossing chords and get the fourth. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Segment length AP
- 3
- Segment length PB
- 4
- Segment length CP
- 2
- Segment length PD
- 6
- Calculated segment
- PD
- Chord-segment product
- 12
Solve the intersecting chords theorem: supply any three of the four segments AP, PB, CP and PD, and the tool returns the missing segment plus the product AP·PB, using the relation AP·PB = CP·PD.
The intersecting chords theorem
When two chords of a circle cross at an interior point P, the products of their divided segments are equal: for chord AB split into AP and PB and chord CD split into CP and PD, the theorem states AP·PB = CP·PD. This calculator inverts that identity. You pass exactly three of the four segment lengths as the fields AP, PB, CP and PD, each a positive number, and it algebraically recovers the one you left out.
Because the relation is a plain multiplicative equality with no trigonometry, the answer is exact up to the rounding you request and never depends on the order in which you list the known segments.
Which segment it solves and how
The tool detects the single omitted field and picks the matching rearrangement of AP·PB = CP·PD. If PD is missing it computes AP·PB/CP; if CP is missing, AP·PB/PD; if PB is missing, CP·PD/AP; and if AP is missing, CP·PD/PB. The recovered length is then rounded half-up to the requested number of decimals, controlled by the optional decimals field, an integer from 0 to 10 that defaults to 6.
The product output AP·PB is rounded the same way, once, from the raw multiplication rather than from an already-rounded segment, so it stays consistent with the resolved values.
Validation and edge cases
You must provide precisely three of AP, PB, PD and CP: passing four leaves nothing to solve and passing two or fewer is under-determined, both rejected with a clear message. The decimals field is bounds-checked to the 0 to 10 range and must be an integer. A degenerate configuration where the divisor segment would force a non-finite result is caught and rejected rather than returning Infinity.
What you can do with it
Geometry homework and proofs
Recover a missing chord segment and verify AP·PB = CP·PD without drawing the circle by hand.
CAD and layout checks
Confirm intersection distances in circular arc constructions from three measured lengths.
FAQ
What must I provide?
Exactly three of the four segments AP, PB, CP and PD, each a positive number. The tool solves the fourth from AP·PB = CP·PD.
How is the missing segment found?
By rearranging AP·PB = CP·PD for the omitted field, then rounding half-up to your decimals (0 to 10, default 6).
What is the product output?
It is AP·PB, the shared value both chord pairs equal, rounded once from the raw multiplication.
What does the intersecting chords theorem say?
When two chords cross inside a circle, the two pieces of one chord multiply to the same value as the two pieces of the other. With pieces of 3 and 4 on one chord and 2 on the other, the last piece must be 6, since both products are 12.
How many decimals does the answer show?
Six by default, and you can choose any whole number from 0 to 10. The rounding goes half up, once, from the raw multiplication, so the product stays consistent with the recovered segment.
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/intersecting-chords \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"AP":3,"PB":4}'const res = await fetch("https://api.kit.forhosting.com/math/intersecting-chords", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"AP": 3,
"PB": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/intersecting-chords",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"AP": 3,
"PB": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/intersecting-chords", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"AP":3,"PB":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"AP":3,"PB":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/intersecting-chords", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"AP": 3,
"PB": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.intersecting_chords",
"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
The intersecting chords theorem
The output echoes all four resolved lengths, names which one was reconstructed in missing_segment, and reports product, the common value AP·PB that both chord pairs must share.
Which segment it solves and how
For the worked example AP=3, PB=4 and CP=2, the missing PD resolves to 6 and product is 12, which confirms that 3·4 equals the product 2·6 for the other chord.
Validation and edge cases
Every supplied segment must be a finite number strictly greater than zero; a zero, negative, non-numeric or non-finite value raises a typed invalid_input error naming the offending field, such as "CP inválido".
Integration use cases
Batch verification via API
Feed many chord configurations as JSON and get the resolved fourth segment and product for each.
Technical questions
What happens with bad input?
Zero, negative, non-finite, or the wrong count of segments returns a typed invalid_input error naming the field.
Is it free?
Yes in the browser as a tier A widget. The deterministic API costs $0.002 per successful call; failed tasks are not charged.