Circle theorems calculator
Enter one angle, pick a circle theorem and get the related angle, so a central angle of 80 degrees gives an inscribed angle of 40 on the same arc. It also covers cyclic quadrilaterals, the semicircle, tangents and the alternate segment. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 80
- Theorem
- Inscribed angle
- Result degrees
- 40
Circle theorems connect angles and arcs in a circle: an inscribed angle is half the central angle on the same arc, opposite angles of a cyclic quadrilateral add up to 180 degrees, an angle in a semicircle is always 90, and the angle between a tangent and a chord equals the angle in the alternate segment.
Inscribed and central angles
The most-used circle theorem relates two ways of looking at the same arc. A central angle has its vertex at the center of the circle and its measure equals the arc it intercepts. An inscribed angle has its vertex on the circumference and intercepts the same arc, and the theorem says it is exactly half the central angle.
So if a central angle of 80 degrees subtends an arc, any inscribed angle subtending that same arc measures 40 degrees — no matter where on the remaining circumference the vertex sits. Because the relationship is an exact factor of two, the result is returned to six decimal places with no rounding surprises.
Angles in semicircles and cyclic quadrilaterals
Two classic corollaries follow directly from the inscribed angle theorem. The cyclic quadrilateral theorem says that when all four vertices of a quadrilateral lie on a circle, each pair of opposite angles is supplementary — they sum to 180 degrees. Given one angle of 110 degrees, the opposite angle must be 70.
These two results look like coincidences until you see they are the same halving rule applied to arcs that together make a whole circle.
Tangents, chords and the alternate segment
Tangents add two more rules that appear constantly in problems. Second, the alternate segment theorem states that the angle between a tangent and a chord through the point of contact equals the inscribed angle in the alternate segment, the part of the circle on the other side of the chord. Given an inscribed angle of 55 degrees in the alternate segment, the tangent-chord angle is also 55 degrees.
These rules let you move information across the boundary of the circle, from outside lines to inside angles and back.
What you can do with it
Solve geometry homework
Given a central angle of 80 degrees, get the inscribed angle of 40 degrees instantly, with the rule stated so you can show your reasoning.
Check a CAD or technical drawing
Verify that an angle inscribed in a semicircle is exactly 90 degrees, or that a tangent meets the radius at a right angle.
Complete a cyclic quadrilateral
From one known angle, compute the supplementary opposite angle to confirm the four vertices really lie on a circle.
FAQ
What units does it use?
Degrees. The input value must be a non-negative finite number and the result is returned in degrees, rounded to six decimal places.
Is the computation deterministic?
Yes. It is pure arithmetic with no randomness, no clock and no network access, so the same input always yields the same output.
How do I find the opposite angle in a cyclic quadrilateral?
Subtract the known angle from 180 degrees. When all four corners lie on a circle, opposite angles add up to 180, so an angle of 110 degrees faces one of 70.
What does the alternate segment theorem say?
The angle between a tangent and a chord at the point of contact equals the inscribed angle on the other side of the chord. If that inscribed angle is 55 degrees, the tangent-chord angle is also 55.
Why is an angle in a semicircle always 90 degrees?
Because the arc is half the circle, so its central angle is 180 degrees and the inscribed angle is half of that. Any triangle drawn from the two ends of a diameter to a third point on the circle has a right angle.
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/circle-theorems \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"theorem":"inscribed_angle","value":80}'const res = await fetch("https://api.kit.forhosting.com/math/circle-theorems", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"theorem": "inscribed_angle",
"value": 80
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/circle-theorems",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"theorem": "inscribed_angle",
"value": 80
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/circle-theorems", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"theorem":"inscribed_angle","value":80}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"theorem":"inscribed_angle","value":80}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/circle-theorems", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"theorem": "inscribed_angle",
"value": 80
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.circle_theorems",
"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 API takes a theorem name and one given angle or arc measure in degrees, and returns the requested angle in a single deterministic call — the same computation you would do on paper, done exactly and instantly, with no geometry software and no guesswork about which rule applies.
Inscribed and central angles
The circle theorems calculator runs this both ways: choose inscribed_angle with a central angle or arc measure to get the inscribed angle, or choose central_angle with an inscribed angle to recover the central angle and the arc. This pair covers the majority of homework and exam problems, and it is also what you need when reconstructing arcs from measured angles in surveying, CAD sketches and astronomy diagrams.
Angles in semicircles and cyclic quadrilaterals
Thales' theorem states that any angle inscribed in a semicircle is a right angle: since the arc is half the circle, the central angle is 180 degrees and the inscribed angle is exactly 90. That is why a triangle drawn from the two ends of a diameter to any third point on the circle is always right-angled. Select angle_in_semicircle or cyclic_quadrilateral and the endpoint applies the rule for you, which is handy when checking a construction, validating that four points are concyclic, or solving for the missing corner of a quadrilateral inscribed in a circle.
Tangents, chords and the alternate segment
First, a tangent line is perpendicular to the radius drawn to the point of tangency, so the tangent_radius theorem always returns 90 degrees — a fact used to locate centers and to build perpendiculars in technical drawing. The API accepts the given measure as a non-negative finite number in degrees and answers deterministically, so the same request always produces the same result — in your browser for free, or via the API at $0.002 per call when you automate it.
Technical questions
What does it cost?
$0.002 per request via the API. It is also free to run in your browser on this page.
Which theorems are supported?
Six: inscribed_angle, central_angle, angle_in_semicircle, cyclic_quadrilateral, tangent_chord and tangent_radius.
Why do angle_in_semicircle and tangent_radius still need a value?
They always return 90 degrees, but the input schema requires a valid non-negative number so malformed requests are rejected consistently.
What happens with invalid input?
The request is rejected with an invalid_input error — for example when the value is negative, not a number, or the theorem name is unknown — and you are not charged.